It is not likely that Pontius Pilate believed Jesus was the Son of the God of Israel. As he was a Roman Officer, he likely believed in a pantheon of gods which, as far as we know, did not include יהוה (YHWH - the God of Israel).
What you must consider is the political climate of the period of time of the Roman occupation of the region of Judea in the second temple period of the first century, the reasons the Jewish leaders of Jesus’ day wanted him dead, and the reasons they involved the authority of the Roman occupation.
In the period, according to Roman law, the Jews were not allowed to execute anyone… but Rome could (and did frequently).
The Jewish Temple leaders (not all the Jews - certainly the first believers of Jesus were all Jews, and there were thousands of them) believed Jesus was guilty of blasphemy - claiming to be the “Son of Man” (a reference to the prophecies of Daniel chapter 7, which is a direct claim to be the Jewish Messiah and the eternal King… according to Psalm 45, God Himself).
If he was correct, they believed they would lose their influence with the people, but that he should be capable of amassing an army — or angels — to take out the Roman occupation — since restoring the Kingdom to Israel is something they believed that the Messiah would ultimately do. If he was incorrect, and really did commit blasphemy, it is possible that this falls under a situation in the Torah that merited being executed (see Leviticus 24:10–23). Either way, “Kill Yeshua” was the goal of the day, and since they couldn’t do it without Rome, they had to involve Pilate. This is spelled out explicitly in John 18:29–31:
So Pilate went outside to them and said, “What accusation do you bring against this man?” They answered him, “If this man were not doing evil, we would not have delivered him over to you.” Pilate said to them, “Take him yourselves and judge him by your own law.” The Jews said to him, “It is not lawful for us to put anyone to death.”
So Rome — Pilate — gets involved.
Now. What would Rome’s interest be? Certainly not in the Torah. It isn’t unlawful according to Roman law to think yourself, or claim yourself to be God, or the Jewish Messiah, or anything else. The question in the mind of Rome, and therefore, in the mind of Pontius Pilate, is, do you want to be King and rule instead of Caesar or instead of the rulers Caesar put in place here.
I want to point out the irony of this. Pilate wouldn’t want Jesus killed for the same reasons the Jewish Temple leaders want Jesus killed at all. Not even in the same ballpark. The Jews wanted him removed for religious reasons and because he was claiming to be the Messiah, but not in way that results in removing Rome by force. Pilate would only be interested in political and military reasons that do include removing Rome by force or otherwise usurping Rome’s authority.
Let me drive this home: If Jesus really would be interested in trying to usurp the Roman occupation, and removing Rome by force, the Jewish Temple leaders probably would have been totally on-board with him (look up Simon ben Kosiba if you don’t believe me — they were totally on-board), but he actually would have been deemed worthy of execution by Pilate.
But that isn’t how it plays out. Let’s watch (I’ll reference several verses in the ESV from John 18).
So again, Pilate’s interest is finding out if Jesus wants to rule instead of Caesar.
He asks that question directly:
So Pilate entered his headquarters again and called Jesus and said to him, “Are you the King of the Jews?”
Jesus wants to know if Pilate is asking because Pilate wants to know if Jesus is the Messiah for himself, or if he is interested in the position Jesus is in under Rome’s rule,and Pilate answers:
Jesus answered, “Do you say this of your own accord, or did others say it to you about me?” Pilate answered, “Am I a Jew? Your own nation and the chief priests have delivered you over to me. What have you done?”
Jesus, understanding now, Pilate is speaking on behalf of Rome and not of himself, answers Rome:
Jesus answered, “My kingdom is not of this world. If my kingdom were of this world, my servants would have been fighting, that I might not be delivered over to the Jews. But my kingdom is not from the world.”
He’s saying “I’m no threat to Rome or Caesar or any nation or human king.” (He’s also saying one other important thing: “My kingdom, unlike the world, doesn’t wage war by fighting.”) Jesus holds the opinion that true followers of the God of Israel (and Himself) would be better citizens and more loyal to their nations/governments and each other because that is what the God of Israel wants, so long as they are loyal to the God of Israel and him first. (See Mark 12:13–17 — he’s saying Caesar can have what he wants as long as God can have you.)
Pilate sees Jesus is not a threat to Rome according to his own testimony, and asks if the Jews are accurate in their claims. This is the next logical step, because if Jesus says no, any good ruler would investigate evidence. Also, Pilate doesn’t trust the Jewish Temple leaders. It’s no secret they are not fond of the Roman occupation.
(Jesus already explained his servants are not fighting. The point would be to see if that is true. Spoiler: this moment is why Jesus told Peter to put back his sword, and healed the ear Peter sliced off in the garden of Gethsemane — see earlier in the same chapter, John 18. In my opinion, it is why that story was included — to drive home Jesus isn’t interested in force, and the only guy actually injured due to a show of force was immediately healed and made whole by Jesus.)
So to regroup… Pilate wants to test the accuracy of the opinion of the Jewish Temple leaders according to Jesus (a known and popular Rabbi and teacher). Jesus answers — I’m paraphrasing to explain what is happening — That’s what YOU say… I have only been saying I have been born and purposed to confirm the truth — if you want truth, listen to me. Here’s what John recorded for us:
Then Pilate said to him, “So you are a king?” Jesus answered, “You say that I am a king. For this purpose I was born and for this purpose I have come into the world—to bear witness to the truth. Everyone who is of the truth listens to my voice.”
Pilate responds confused, not as to why this man is offensive to the Jews — he doesn’t seem to care — but why they would think Rome would be offended by him. Or to put it another way, why do they think _I_ should deal with this?
Pilate said to him, “What is truth?”
This is like saying “What is that small group with 2 butter knives over there compared to the armies of Rome?” Modern translation: “Truth is no threat to Rome.”
After he had said this, he went back outside to the Jews and told them, “I find no guilt in him.”
Did Pilate believe? We have no indication from the Gospels that he did. Simply that he didn’t think Jesus was a threat to Rome, and was thereby innocent of the crime that would have actually merited his crucifixion.