Even if Matthew's gospel account was written initially in Hebrew, there are no autographs of any of the books of the Bible, so we could never know. The oldest available manuscript copies of Matthew are all in Greek. But that does not prove he wrote for Greek-speaking Jews, or Greek-speaking Gentiles. Greek was the language of commerce, it was used for all legal matters, and politics. Perhaps an illustration could be that in Europe there are many different languages used, but to facilitate commercial, legal and political and educational matters, English is the common language used throughout Europe. So in the first century; koine Greek was written.
We are therefore no further forward as to Matthew's intended audience until we start examining what he wrote. He starts with a genealogy that is entirely comprised of Jewish names, starting with the patriarch of Israel, Abraham, and including kings of Israel. Clearly, Jewish people would be keenly interested in such a connection. One of the opening verses gives another clue. The angel said (of the naming of Mary's child), "You shall call his name Jesus, for he shall save his people from their sins" (1:21): 'his people' being the Jewish people. So, in the first chapter, the attention of Jewish readers is particularly 'signaled'.
It is not until chapter 12 that Matthew makes any direct reference (in the sense of 'appeal') to non-Jewish readers. He does this by quoting a Hebrew prophecy about Messiah's name being what the nations (the Gentiles) will put their hope in (12:21 quoting Isaiah 42:1-4).
However, Matthew is unique, in all the gospel accounts, in mentioning the word 'ecclesia' - Church. He uses that word three times, yet it is never used in any of the other three gospels. This surely indicates that those people who knew themselves to belong to the Church that Jesus builds (Mat. 16:18) would see significance in this. By the time Matthew wrote his gospel, the initial preponderance of Jewish members of the ecclesia had moved over to a Gentile one. The word is used twice more in 18:17. Here Jesus is giving instructions on handling problems between members of the Church. Surely Matthew's unique mention of the Church shows that he was aiming his gospel account at those where were members of it? And those members were, by then, a mixture of Jews and Gentiles.
Supporting this, Matthew concludes his account by depicting the risen Christ appearing beyond the pale of Israel, looking out upon all nations from the heights of that appointed mountain in Galilee of the Gentiles, there to command his disciples. Those already members of his Church are told to "go and make disciples of all nations", showing that the Church was inclusive of Gentiles.
So, although the opening chapter begins with the Jews, the last chapter has the Gentiles included. And, throughout, the account is instruction to all who are members of the Church Christ is building, both Jews and Gentiles.