Q. Could you please tell me what happened between Paul and John Mark that made Paul angry with him? Does this establish a Biblical way of dealing with disagreement?
I believe you will find your answer in the following verses:
37 Barnabas wanted to take John, called Mark, along with them also. 38 But Paul kept insisting that they should not take him along who had deserted them in Pamphylia and had not gone with them to the work. 39 And there occurred such a sharp disagreement that they separated from one another, and Barnabas took Mark with him and sailed away to Cyprus.
I believe that verse 38 answers your question. Apparently, John Mark left them in Pamphylia, and returned home. We don't know why, and we know nothing of the circumstances. Paul's judgment was that John Mark should not have left them, Barnabas did not judge him as harshly. The good news is that sometime later Paul wanted John Mark to assist them in their mission work.
2 Timothy 4:11
"Only Luke is with me. Pick up Mark and bring him with you, for he is useful to me for service."
The Biblical way of dealing with problems is found in Matthew 18:15-17
15* "If your brother sins, go and show him his fault in private; if he listens to you, you have won your brother. 16* "But if he does not listen to you, take one or two more with you, so that BY THE MOUTH OF TWO OR THREE WITNESSES EVERY FACT MAY BE CONFIRMED. 17* "If he refuses to listen to them, tell it to the church; and if he refuses to listen even to the church, let him be to you as a Gentile and a tax collector.