No, Jesus, Mary, and Joseph were Israelites of the tribe of Judah and the House of David.
I didn't see anything in the Wikipedia article saying that it was predominately Arab during the time of Christ. As far as I'm aware, Nazareth was predominately inhabited by Galilean Judahites/Israelites ("Jєωs" would be an incorrect term, as modern "Jєωs" are Edomites, rather than Israelites).
If I recall correctly, the Arabs (Ishmaelites) were living in what is now Saudi Arabia, and later, after the rise of Islam, started settling in Palestine.