Yes — lime is safe but very acidic, so it's an occasional novelty in tiny amounts, like lemon.
Keeper's noteA flake of lime flesh now and then is harmless fun. Remove seeds, and skip the peel beyond an occasional zest.
What to watch out for with lime
Lime is non-toxic and vitamin-C-rich, but it's among the most acidic citrus fruits, which can upset a bird's stomach in any real quantity. There's no nutritional need for it, so treat it as a curiosity for birds that enjoy sour flavors.