Since sound doesn't play when you pick up an invulnerability power-up in Quake/Doom, I don't think there's a sound file that I can change. So this may require a little modding.
You sure? There's one for picking it up, and there's more sound for it wearing off. Same as using a quad in quake. Surely those would be isolated sound files. Although I can't remember what its like playing online but, because I think you can pick up and use later in that case, in which case it would be only when used, not when picked up.
Just tested it, so for example with Quake 1, go open up pak0.pak, check out the sound files. damage.wav's are for quad, inv.wav's are for invulnerability, and yeah it got the use and wearing off sounds as well. Not gonna try replacing them, I'm happy with my sounds.