Q: How do you know that Luke was probably Jewish or a Jewish convert?
A: The key verse that must be taken into account here is Romans 3:1-2, which says that unto the Jews were commited the oracles of God. The “oracles” refer to written Scripture, and it clearly states that the Jews were the ones who would write Scripture. Furthermore, the Gospel of Luke and the Book of Acts in the original Greek language contain many “Hebrewisms,” meaning that though the writer is writing Greek, he is using a Hebrew word order. Basically, he is writing the kind of Greek written by Greek-speaking Jews rather than Greek-speaking Gentiles.
Furthermore, he also writes in a Jewish way or style. For example, he is very careful to write that when people go to Jerusalem, it is always “up” to Jerusalem; and when they leave Jerusalem, it is always going “down” from Jerusalem. These are the ways in which Jews write, as opposed to Gentiles.
Actually, your questions should really be asked in a different way: How do we know that Luke was a Gentile? People often point out that his name is a Gentile name, which is true. However, the name “Paul” is a Gentile name as well, but no one questions Paul's Jewishness.