A large slab of Penrose's book The Emperor's New Mind is, if I recall correctly, devoted to describing how certain structures of the brain just might have some quantum mechanics involved. I don't recall the details and I don't know whether it's all been disproven as we understand the brain better.
I really have to read this book, since I've heard this sort of thing before and it never made sense to me why this matters. I mean, of course quantum mechanics is involved in the operations of the brain; it's involved in the operations of literally every single aspect of physical reality except gravity. People say things like "the brain uses quantum mechanics" and I think "yes, but so does Coca Cola, erosion, and rust."
I think part of it is conflating a few different views:
1.) The brain is some sort of quantum computer (almost certainly false)
2.) The brain as a physical system is somehow sensitive to quantum uncertainty, for example, in the probability of a neuron firing when stimulated. We might hypothesize that this explains our subjective experience of free will: we might be able to chose our world in the sense of the many worlds interpretation, but the worlds we're limited to are just different patterns of deterministic activity within the brain. This is very speculative, but appealing, and I think it deserves more consideration.
Quantum biology is making inroads in a few areas, such as in the mechanism of chlorophyll. I would not be surprised if it starts popping up in bits of neurology, even in the parts that we currently understand classically.
There's a huge difference between molecule-scale quantum chemical phenomena (as in photosynthesis) and quantum computation. Finding the former in the brain would be very cool, but it wouldn't be terribly surprising or revolutionize cognitive science the way the latter would.
There is no quantum level magic.