One definitely could draw a contrast between the two men, having similar statements. However, the statements do not mean the same thing. If anything, they complement and progress.
Pilate questions the topic of truth in a politically volatile environment where every claim is filled with propaganda and undisclosed agendas. Judea was under the overlapping rule of two legislative bodies of both the Sanhedrin and the Roman Senate, along with Herod and Pilate with their own overlapping fiat powers. Pilate answered to Caesar Tiberius who often killed people just for fun while Herod pretty much answered to no one.
By contrast, Pharaoh had absolute political sovereignty and questioned the existence or nature of God Himself.
We might also consider that Pharaoh had the dominant power over an enslaved nation. Pilate faced political threat of unrest from the Jewish leaders to provoke the wrath of Tiberius. The powers of Pharaoh and Pilate were very different.
We see how politically charged the environment was, and that Jesus was doing redemptive work even while Pilate oversaw his execution...
Luke 23:12 (NASB emphasis added)
Now Herod and Pilate became friends with one another that very day; for before they had been enemies with each other.
Explanation of "What is truth?"
Pilate also makes a public statement, claiming Jesus as "King of the Jews", refusing to take it down when opposed—because he was opposed about another factual claim to the question "What is truth?".
John 19:29-22 (NASB)
19 Pilate also wrote an inscription and put it on the cross. It was written, “JESUS THE NAZARENE, THE KING OF THE JEWS.” 20 Therefore many of the Jews read this inscription, for the place where Jesus was crucified was near the city; and it was written in Hebrew, Latin and in Greek. 21 So the chief priests of the Jews were saying to Pilate, “Do not write, ‘The King of the Jews’; but that He said, ‘I am King of the Jews.’” 22 Pilate answered, “What I have written I have written.”
V21 itself is a debate about truth stated in the public record and Pilate's struggle with it, directly from the story. That explains best what Pilate probably meant.
What is the relationship?
Can a parallel be drawn between Pilate's statement and Pharaoh's of Exodus?
Certainly, but it would be a contrast because in the end, the truth that Pilate claims is that "The LORD" is Jesus of Nazareth. Pharaoh asks the question in defiance; Pilate answers it as a public confession of faith in the man he tried to save, but was forced to execute with a knife to his back.