Was Pilate also the father of the son who was healed by Jesus in John 4? I am new to Christianity, but I have read the Gospel of John and have watched a movie about Jesus. Was the Pilate in John 18:28 the same man whose son Jesus healed in John 4:46?
No. The Pilate of John 18:28 was the Roman procurator of Judea in Jesus' day. Pilate was the civil authority who thought he had the power to turn Jesus over to the Jews to be crucified. A procurator was any of various imperial officials with fiscal or administrative powers. The equivalent today might be a governor.

Regarding the "nobleman" in John 4:46-54, from the NET Notes comes the following:


  Although βασιλικός (basiliko) has often been translated “nobleman” it almost certainly refers here to a servant of Herod [Antipas], tetrarch of Galilee (who in the NT is called a king, Matt 14:9, Mark 6:14-29). Capernaum [the town where the sick son was] was a border town, so doubtless there were many administrative officials in residence there [, and this unnamed nobleman was one of them.]


Jesus healed this man's son "long distance," so to speak, because He knew this nobleman had the faith to believe Jesus would do as He said He would. There was no need for Jesus to leave what He was doing and walk about 20 miles or so from where He was (viz., Cana) to Capernaum, where the sick boy was. 

In our day, Pilate would be the equivalent of the governor of, say, Pennsylvania. The nobleman would be the equivalent of the Director of the Pennsylvania Department of Revenue and Taxation; in other words, a high-level bureaucrat. 

If on the other hand the nobleman was a military man (and this is a distinct possibility), he would be, approximately, the equivalent of a lieutenant, major, or even a colonel, whereas Pilate would be a general.