Acts 4:13 says, "13 Now when they saw the boldness of Peter and John, and perceived that they were unlearned and ignorant men, they marvelled; and they took knowledge of them, that they had been with Jesus."  This does not say that the high priest did not know John.  It simply states that they noted the boldness of Peter and John; and that they had not been through their schools; and that they had been with Jesus.

If you read through the whole Book of John, it seems evident that the disciple in question is John.

John 19:25-27 says, “25 Now there stood by the cross of Jesus his mother, and his mother's sister, Mary the wife of Cleophas, and Mary Magdalene.   26 When Jesus therefore saw his mother, and the disciple standing by, whom he loved, he saith unto his mother, Woman, behold thy son!  27 Then saith he to the disciple, Behold thy mother! And from that hour that disciple took her unto his own home.”    

John 20:1-8 says, “1  The first day of the week cometh Mary Magdalene early, when it was yet dark, unto the sepulchre, and seeth the stone taken away from the sepulchre.  2 Then she runneth, and cometh to Simon Peter, and to the other disciple, whom Jesus loved, and saith unto them, They have taken away the Lord out of the sepulchre, and we know not where they have laid him.  3 Peter therefore went forth, and that other disciple, and came to the sepulchre.  4 So they ran both together: and the other disciple did outrun Peter, and came first to the sepulchre.  5 And he stooping down, and looking in, saw the linen clothes lying; yet went he not in.  6 Then cometh Simon Peter following him, and went into the sepulchre, and seeth the linen clothes lie,  7 And the napkin, that was about his head, not lying with the linen clothes, but wrapped together in a place by itself.  8 Then went in also that other disciple, which came first to the sepulchre, and he saw, and believed.”

John 21:5-7 says, “5 Then Jesus saith unto them, Children, have ye any meat? They answered him, No.   6 And he said unto them, Cast the net on the right side of the ship, and ye shall find. They cast therefore, and now they were not able to draw it for the multitude of fishes.  7 Therefore that disciple whom Jesus loved saith unto Peter, It is the Lord. Now when Simon Peter heard that it was the Lord, he girt his fisher's coat unto him, (for he was naked,) and did cast himself into the sea.”

John 21:18-24 says, “18 Verily, verily, I say unto thee, When thou wast young, thou girdedst thyself, and walkedst whither thou wouldest: but when thou shalt be old, thou shalt stretch forth thy hands, and another shall gird thee, and carry thee whither thou wouldest not.  19  This spake he, signifying by what death he should glorify God. And when he had spoken this, he saith unto him, Follow me.  20 Then Peter, turning about, seeth the disciple whom Jesus loved following; which also leaned on his breast at supper, and said, Lord, which is he that betrayeth thee?  21 Peter seeing him saith to Jesus, Lord, and what shall this man do?  22 Jesus saith unto him, If I will that he tarry till I come, what is that to thee? follow thou me.  23 Then went this saying abroad among the brethren, that that disciple should not die: yet Jesus said not unto him, He shall not die; but, If I will that he tarry till I come, what is that to thee?  24 This is the disciple which testifieth of these things, and wrote these things: and we know that his testimony is true.”

John had very clearly learned a lesson from an earlier incident in his life.  Matthew 20:20-28 says, “20 Then came to him the mother of Zebedee's children with her sons, worshipping him, and desiring a certain thing of him.  21 And he said unto her, What wilt thou? She saith unto him, Grant that these my two sons may sit, the one on thy right hand, and the other on the left, in thy kingdom.  22 But Jesus answered and said, Ye know not what ye ask. Are ye able to drink of the cup that I shall drink of, and to be baptized with the baptism that I am baptized with? They say unto him, We are able.  23 And he saith unto them, Ye shall drink indeed of my cup, and be baptized with the baptism that I am baptized with: but to sit on my right hand, and on my left, is not mine to give, but it shall be given to them for whom it is prepared of my Father.  24 And when the ten heard it, they were moved with indignation against the two brethren.  25 But Jesus called them unto him, and said, Ye know that the princes of the Gentiles exercise dominion over them, and they that are great exercise authority upon them.  26 But it shall not be so among you: but whosoever will be great among you, let him be your minister;  27 And whosoever will be chief among you, let him be your servant:  28 Even as the Son of man came not to be ministered unto, but to minister, and to give his life a ransom for many."

John learned this lesson well, and was very humble in his writing when God used him to write the Book of John.

John 18:14-16 says, “14 Now Caiaphas was he, which gave counsel to the Jews, that it was expedient that one man should die for the people.  15 And Simon Peter followed Jesus, and so did another disciple: that disciple was known unto the high priest, and went in with Jesus into the palace of the high priest.  16 But Peter stood at the door without. Then went out that other disciple, which was known unto the high priest, and spake unto her that kept the door, and brought in Peter."

This does not contradict Acts 4:1-13: “1 And as they spake unto the people, the priests, and the captain of the temple, and the Sadducees, came upon them,   2 Being grieved that they taught the people, and preached through Jesus the resurrection from the dead.  3 And they laid hands on them, and put them in hold unto the next day: for it was now eventide.  4 Howbeit many of them which heard the word believed; and the number of the men was about five thousand.  5 And it came to pass on the morrow, that their rulers, and elders, and scribes,  6 And Annas the high priest, and Caiaphas, and John, and Alexander, and as many as were of the kindred of the high priest, were gathered together at Jerusalem.  7 And when they had set them in the midst, they asked, By what power, or by what name, have ye done this?  8 Then Peter, filled with the Holy Ghost, said unto them, Ye rulers of the people, and elders of Israel,  9 If we this day be examined of the good deed done to the impotent man, by what means he is made whole;  10 Be it known unto you all, and to all the people of Israel, that by the name of Jesus Christ of Nazareth, whom ye crucified, whom God raised from the dead, even by him doth this man stand here before you whole.  11 This is the stone which was set at nought of you builders, which is become the head of the corner.  12 Neither is there salvation in any other: for there is none other name under heaven given among men, whereby we must be saved.  13 Now when they saw the boldness of Peter and John, and perceived that they were unlearned and ignorant men, they marvelled; and they took knowledge of them, that they had been with Jesus.”

This passage is not saying that Annas and Caiaphas did not know John.  What they noted was the boldness of Peter and John, and that they were “unlearned and ignorant men,” that is, and they were not men who had been trained in their schools.  They understood that Peter and John had been with Jesus, that is, that they had spent time with Jesus and were trained by Him.  There are no personal words recorded here between John and the high priest, but that should not seem strange, because the Jews immediately turned on anyone who confessed that Jesus was the Christ.

John 9:18-34 says, “18 But the Jews did not believe concerning him, that he had been blind, and received his sight, until they called the parents of him that had received his sight.  19 And they asked them, saying, Is this your son, who ye say was born blind? how then doth he now see?  20 His parents answered them and said, We know that this is our son, and that he was born blind:  21 But by what means he now seeth, we know not; or who hath opened his eyes, we know not: he is of age; ask him: he shall speak for himself.  22 These words spake his parents, because they feared the Jews: for the Jews had agreed already, that if any man did confess that he was Christ, he should be put out of the synagogue.  23 Therefore said his parents, He is of age; ask him.  24 Then again called they the man that was blind, and said unto him, Give God the praise: we know that this man is a sinner.  25  He answered and said, Whether he be a sinner or no, I know not: one thing I know, that, whereas I was blind, now I see.  26 Then said they to him again, What did he to thee? how opened he thine eyes?  27 He answered them, I have told you already, and ye did not hear: wherefore would ye hear it again? will ye also be his disciples?  28 Then they reviled him, and said, Thou art his disciple; but we are Moses' disciples.  29 We know that God spake unto Moses: as for this fellow, we know not from whence he is.  30 The man answered and said unto them, Why herein is a marvellous thing, that ye know not from whence he is, and yet he hath opened mine eyes.  31 Now we know that God heareth not sinners: but if any man be a worshipper of God, and doeth his will, him he heareth.  32 Since the world began was it not heard that any man opened the eyes of one that was born blind.  33 If this man were not of God, he could do nothing.  34 They answered and said unto him, Thou wast altogether born in sins, and dost thou teach us? And they cast him out."

Saul, himself, was known to the high priest.  Acts 9:1-2 says, “1 And Saul, yet breathing out threatenings and slaughter against the disciples of the Lord, went unto the high priest,  2 And desired of him letters to Damascus to the synagogues, that if he found any of this way, whether they were men or women, he might bring them bound unto Jerusalem.”  Saul got saved, and he was then a target of hatred by the Jews.  Acts 9:19-30 says, “19 And when he had received meat, he was strengthened. Then was Saul certain days with the disciples which were at Damascus.  20 And straightway he preached Christ in the synagogues, that he is the Son of God.  21 But all that heard him were amazed, and said; Is not this he that destroyed them which called on this name in Jerusalem, and came hither for that intent, that he might bring them bound unto the chief priests?  22 But Saul increased the more in strength, and confounded the Jews which dwelt at Damascus, proving that this is very Christ.  23 And after that many days were fulfilled, the Jews took counsel to kill him:  24 But their laying await was known of Saul. And they watched the gates day and night to kill him.  25 Then the disciples took him by night, and let him down by the wall in a basket.  26 And when Saul was come to Jerusalem, he assayed to join himself to the disciples: but they were all afraid of him, and believed not that he was a disciple.  27 But Barnabas took him, and brought him to the apostles, and declared unto them how he had seen the Lord in the way, and that he had spoken to him, and how he had preached boldly at Damascus in the name of Jesus.  28 And he was with them coming in and going out at Jerusalem.  29 And he spake boldly in the name of the Lord Jesus, and disputed against the Grecians: but they went about to slay him.  30 Which when the brethren knew, they brought him down to Caesarea, and sent him forth to Tarsus.”  

Old friendships meant nothing to the Jews, when it came to someone accepting Jesus as the Messiah.  They were ready to kill anyone who took that position.