Translation of the Greek New Testament began early on in the history of Christianity. Apparently no translations were made into Hebrew. Why?

To expand on Kenneth Bakke’s answer: the Jewish Christians who were the original Christian movement appear to have survived into the 5th century. We don’t know that much about them, but we do know that they had their own Gospels (Jewish–Christian gospels). They certainly would not have had any time for the Gospel of John, as the Gospel that most overtly makes a God of Jesus, and they may have been reluctant to take on Matthew or Luke, with their Christology and anti-Jewish sentiment.

That said, of the three known Jewish Christian gospels (as tentatively reconstructed from patristic citations), the Gospel of the Ebionites was a Gospel harmony based on the Synoptic Gospels, written in Greek; and the Gospel of the Nazarenes was based on Matthew and written in Aramaic. (The Gospel of the Hebrews, which explicitly denied the divinity of Jesus and of the crucifixion, and repudiated Paul, appears to have been an original composition in Greek.) That indicates that there was some translating from Greek into Aramaic; it also indicates that many Jewish Christians could read Greek.

Hebrew was by then a liturgical language; if you wanted to reach an audience outside the synagogue, you would use Aramaic or Greek instead of Hebrew anyway.