There's evidence that biology takes advantage of quantum effects on all levels - all the way from individual chemical bonds and interactions of molecules (quantum biochemistry) up to cellular and multi-cellular. Mostly because there's no way it could work on such small scales and be so energy efficient if it didn't.

So one thing is certain - the brain does use quantum mechanics just like the rest of the body, because otherwise it wouldn't be possible to have so much done inside such a small volume, with such small amounts of energy.

Of course this question is actually meant to be "is brain a quantum computer?" and we don't have any idea.