Was John the Beloved and John Mark the same person?

No. They are two different people. John the apostle was a young disciple of Jesus and formerly a fisherman with James both being sons of Zebedee.

John Mark, the author of the Gospel of Mark was a traveling companion of Paul for a time as referenced in Acts 12:12 and 25; Acts 15:37 and 39 and 2 Timothy 4:11.

The apostle John wrote the Gospel of John while John Mark wrote the Gospel of Mark. The two Gospels are themselves quite different in their style and arrangement of subject matter, bearing witness to two quite different personalities at work under God's inspiration.

The two are quite distinct as John the Apostle is not recorded as having accompanied Paul on any of his journeys, nor is it recorded that John Mark was present as a disciple. One possible reference to John Mark is in Mark 14:51-52 which has no significance except as it were an incidental reference of Mark to himself and his own embarrasment.

This incident if correctly interpreted (as it may well be in the light of later developments where Mark is recorded as having 'left' the missionary work of Paul) demonstrates somewhat of a fearful nature on the part of Mark. John, on the other hand, even was present at Jesus' trial.