Rather than speculating about Jews, it may be more useful to compare Lebanese communities, which we have good studies about, like here.
Comparing Maronites (presumably more representative of the pre-Arab Levant than most populations) and Lebanese Muslims (presumably having some Arab influence) gives the following haplogroup shifts:
Maronite->Muslim
E1b: +5%
G: +3% (!)
I: -2%
J1: +3%
J2: -8%
L: -2%
R: 0%
T: 0%
I don't think I've seen evidence that the ancient Levant had substantial G. I'd continue with the "expansion within the diaspora" hypothesis, personally.
The Maronites are only one small community in the Levant. The data I cited for the Levant includes all the studies I know for Syria, Lebanon, Palestine and Jordan. The average for hg G is 3.5% for both Muslims and Christians.
We will see what proportion of G turns up in future ancient DNA samples from the Near East.