Are you asking whether he explicitly used the title, "Son of God"? If so, it's hard to say. What seems clear is that Jesus claimed (or portrayed himself as being) a fulfillment of the expectation that somebody would sit on the throne of David and usher in all the promises God made. So Jesus was, essentially, claiming to be "The king of the Jews."

This is evident in a number of things, not least of which is the fact that all four gospels say that a sign was placed above Jesus head saying that he was the king of the Jews.

"Messiah" is just one possible title this eschatological character could be referred to as. "Messiah" comes from the Hebrew word for "anointed," and "Christ" comes from the Greek word for "anointed," so they both mean the same thing. Prophets, Priests, and Kings were all called "anointed," but in the New Testament, the term appears to be used primarily in the sense of being the Davidic king.

Jesus was called "Christ" in the earliest layers of our written tradition. The earliest comes in the form of an oral tradition quoted by Paul in 1 Corinthians 15 in which Paul says, "Christ died for sins. . ." Notice that the tradition, which predates Paul, doesn't say "Jesus," or even "Jesus Christ" or "Christ Jesus." It just says, "Christ." This is significant because before you would automatically know who "Christ" referred to, the title would have to be used of a specific person for a while. This pushes the use of the title even earlier than the oral tradition itself.

So that's another reason to think the title (or the office) originated with Jesus himself. A final reason is just that there needs to be a sufficient explanation for why Pilate crucified Jesus. Romans didn't crucify people for just any reason. It was mainly a way fo maintaining the Pax Romana (i.e. the peace of Rome). It was primarily used for people who lead revolutions or caused riots. A would-be king would fit this bill, especially if he had followers.

There are a couple of incidents recorded in the gospels that, if true, would explain what got Jesus arrested and condemned in the first place. The first was that he supposedly rode into Jerusalem on a donkey, consciously acting out a prophecy in Zachariah 9:9 about a king. He was met by people showing, "Hosanna to the son of David." Some scholars doubt this incident happened because it's such an overt messianic claim that if it had actually happened, Jesus would've been arrested and crucified a lot sooner.

Another incident was when he turned over the tables in the Temple area and caused a scene. The gospels are a little bit ambiguous about what came next. One gospel says he predicted the destruction of the Temple. One says that if you destroy the Temple, he will rebuilt it. One gospels says that during his interrogation before the Sanhedrin, somebody claimed that Jesus threatened to destroy the Temple himself.

So it seems that Jesus must've said something about the destruction and/or rebuilding of the Temple, which also is a messianic expectation coming from prophecies like Zechariah 6:12. This would also explain why Jesus got arrested and crucified.

There's also the fact that one of the first things out of Jesus' mouth when he started his minstry was that "the kingdom of God is at hand," and nearly all of his parables are about the "kingdom of God" or the "kingdom of heaven." That, combined with the fact that he chose twelve apostles to be in his inner circle, indicates a messianic claim. Twelve has symbolic meaning since there were originally twelves tribes of Israel. This symbolism is made explicit in Matthew 19:28 and Luke 22:30 where it says the twelves will sit on twelve thrones judging the twelve tribes of Israel. After Solomon died, the David kingdom was split in two with the kingdom of Israel in the north and the kingdom of Judah in the south. The prophets (e.g. Ezekiel 37) predicted that the coming of the messiah would be associated with the reunion of Judah and Israel and an ingathering of all of those who had been scattered among the nations. That's one reason to think Jesus' calling of twelves apostles was a symbolic act indicating his claim to be the messiah. Moreover, since Jesus saw himself as being an authority over the twelves people who represent the twelve tribes shows that Jesus saw himself as being something like a king.

"Messiah" isn't the only title used of the eschatological Davidic king, though. Other titles include "Son of David," "Branch of Jesse," or just "Branch," and even "Son of God." These are all messianic titles. It's unclear whether Jesus explicitly used any of them about himself, but it does seem clear that Jesus claimed to be this person or at least lead his followers to think he was claiming to be this person by his actions and teachings.