Does the New Testament refer to the name Palestine?
There are no references to this name anywhere in the New Testament.
Could Jesus have called Israel by any other name?
In their writings that cover many subjects and historical periods, these historians and travelers occasionally use the name Palestine, usually in the context of Syria and its southern environs: Herodotus (c. 485-420? BC); Josephus (c. 37 to post-100 AD), a Jewish historian; Suetonius (69/75 to post-135 AD); Arrian (c. 90-180? AD) (search here); Appian (c. 95 to post-163 AD); Pausanias (second century AD).
Referencing these historians and travelers is not to say that Jesus (or the New Testament authors) read them, particularly the ones who lived after Jesus and the authors!
Therefore, Jesus had the choice of the name Palestine or Israel. But he chose to remain within the Biblical tradition, calling his country Israel. The New Testament authors also chose the Biblical tradition, exclusively.