What? Absolutely not. Never!
Never bootleg ground
The attraction of bootlegging ground to a GFCI is then, plug-in GFCI testers will work. They're not supposed to work on an ungrounded GFCI. He is doing that because he is a simpleton, or because he must impress an inspector who is a simpleton.
If the GFCI has no ground, it should be labeled No Equipment Ground. And if you stick a 3-light+GFCI tester in there, you should get 1 light. Pushing the test button should do nothing. If you get 2 lights, or a trip, violation.
Why shouldn't a GFCI tester work on a receptacle with only two wires going to it? Because a normal and healthy circuit needs 2 wires. A ground fault is by definition current taking a third path. If there are only 2 wires, the tester can't create a round fault, as no third path is available.
If you really, really want to make a GFCI tester work in an ungrounded GFCI, just get a 2/3 prong cheater with a little ground pigtail and use a partial reel of THHN to extend that down to the basement or anywhere you can get a real ground. That should give you the third path for fault/test current.
Before you retrofit grounds...
2014 Code revisions give you broad freedom to retrofit ground any way practical. Any circuits from the same subpanel can share ground paths as long as they're the required size. But before you order a bunch of drywall spackle... Remember the fundamentals: Current travels in loops, and grounds are concerned with two loops.
First service - hot wire - equipment - neutral - service. The GFCI provides flawless protection for that loop - better than you can have with a ground alone! Which is why they're requiring GFCI in places that already have grounds. So for electrocution protection, the GFCI has covered it - you don't really need a ground.
Second earth - sky - lightning - earth. GFCI won't do a darn thing about that, but that is also not a big issue in an indoor receptacle. It might be worth retrofitting ground if you are worried about earth - carpet - fingers - delicate equipment - earth loops.