No, Jesus would not call it by any other name no matter who ruled over it through the centuries, it was always called Judea. Jesus called it Judea because the territory has always been called that until the recent Palistinian-Israeli conflict:
'The mountains of Judea are first named in the Book of Joshua, in the account of the conquering of Canaan by the Israelites during the creation of the Land of Israel. From that time to the present, more than 3,000 years, the name Judea has been consistently used to describe the territory from Jerusalem south along the Judean mountain ridge line, extending east from the mountains down to the Dead Sea.
Judea and Samaria have been known by these names for unbroken centuries, and were registered as such on official documents and maps, by international institutions and in authoritative reference books right up to about 1950. When the correct names became a problem for Palestinian Arabs trying to make their newly-minted claim on the land, it somehow became 'politically correct' to use 'West Bank' or 'occupied territories' instead of the historically accurate names Judea and Samaria.'
So, those living in the territories of 'biblical Judea' can certainly "flee to the mountains" at the fulfillment of the prophecy and Christ's prophetic words hold true.