Can the 5/8 wave antenna be matched with a tuning stub such as the
J-Pole uses?...

Probably, but seems easier to to use a 5 turn coil at
the base...But I agree...A 5/8 GP is not a very good 2m
antenna. Better off building a high quality 1/4 wave GP.
Seriously... By high quality, I mean instead of 3-4 sloped
radials, use 6-8 radials, and even better , use a 2nd set
for decoupling. You can also build "sleeve" versions.
5/8's are ok on HF, where the angles are not critical, and on
10m, will usually be the best choice. But on 2m, they are usually
lame.
But, don't take my word. Build a 5/8 GP, and then build a 1/4
GP, and see which is best. I bet the 1/4 wave wins. For a
5/8 to work well on VHF/UHF, it needs to be a collinear dual
5/8 design, or at the least, use sloped 5/8 or 3/4 wave radials.
5/8's with 1/4 wave radials have pretty lousy patterns for 2m use.
MK