What kind of surgeon performs fingernail surgery?

A manicurist??? LOL

Seriously though, I guess plastics or a hand surgeon...

a Podiatrist?

A hand surgeon.
I've seen toenail surgery done by a podiatrist. It took about 10 minutes.

If a podiatrist does ingrown toenails then I suppose a hand surgeon would do ingrown fingernails. That would be my guess

We have a couple family docs who come in to do the occasional ingrown toenail. I've never seen a plastic surgeon do an ingrown fingernail, but I suppose they could. I would think that a family doc could do it in their office though!