Pilate was Roman. If at any point (before or after Jesus’ death) he accepted Christ as the Messiah of all peoples and attempted to make amends, then he could have been forgiven. This isn’t for us to truly know.
There are apocryphal writings such as the Gospel of Peter wherein Pilate claims that his “hands were clean” of Christ’s blood. There are also varying anecdotes about his untimely end.
We can speculate all day as to the fate of ancient persons regarding what is “in God’s eyes,” but we are not truly privy to His thoughts, unless revealed to us in the Canon of Scripture. Similarly, we simply have no confirmation of any conversion of heart or lifestyle for Pilate.
I know this isn’t much of an answer, and inasmuch as I hate to recommend Wikipedia, you can look there and use it as a springboard for further research on the life of Pilate.