Physicist: The extremely short, smart-ass answer to this is: of course! Ultimately everything in the universe is built out of tiny quantum things and ultimately everything obeys quantum mechanical laws. But that’s not really the spirit of the question, and it doesn’t tell you anything useful. You could just as easily say that computers take advantage of the same principles behind lightning (electricity), but that doesn’t tell you anything about computers or lightning.
The “connection” between quantum physics and consciousness is pretty famous these days, what with the Secret, Chopra, and stuff like that. Unfortunately, in legit science we don’t have a real, solid definition for consciousness, which makes talking about it a little tricky. To a physicist, consciousness is like obscenity; you know it when you see it. So, while I can’t speak to what consciousness is, I can say it doesn’t seem to have any influence on quantum phenomena (at least, outside of the head). The brain, on the other hand, can be defined (you can even point at it!), and the brain definitely involves quantum mechanics. Chemistry, which is arguably the most important part of biochemistry, is all about quantum mechanics.
But are more interesting quantum effects, like entanglement, quantum teleportation, and quantum computation used by the brain? It seems unlikely, since quantum experiments done in the lab are generally done in very, very carefully controlled, usually cold, environments, involving just a few atoms at a time. By contrast, brains involve many atoms (like… dozens) and are generally warm, and squishy, and very un-lab-like. Any zombie scientist will say the same. And yet the surprising, somewhat hesitant, answer is a resounding: maybe! Recently it’s been shown that many (possibly all) plants use a form of “quantum search”
So far, photosynthesis is the best example of coherent quantum phenomena (stuff involving entangled or controlled states across multiple atoms) in biology, but it’s exciting enough to give rise to the new field of “quantum biology”.
Generally speaking, “coherent states” (which are the kind of nice, clean quantum states you need to have entanglement or any of the other weirder quantum effects) get broken apart in environments as noisy as a biological system. But there are some very slick tricks being used by physicists to over-come noise, like topological quantum computation, quantum error-correcting codes, and robust states (clever combinations of states that are more stable than their constituent states).
It may be that, like chloroplasts in plants, our brains have some cute tricks for maintaining coherent states and possibly computing with them. Weirder things have happened (maybe not a lot weirder). For example; among the many other things they do, ours eyes are continuously running an edge-detection algorithm and our ears preform a physical Fourier transform. So far we’ve only scratched the surface of what kind of tricks our brains use, literally. It’s tricky to study the inside of a brain while keeping it alive, so we know a lot about the eyes, ears, nose, and peripheral nervous system, but surprisingly little about how that information is processed in the brain.
There’s a subtle but important difference between quantum mechanics and magic. So, even if our brains are capable of handling quantum information on a large scale, we can’t expect to have any cool powers because of it. The brain might use entanglement to correlate things on different sides of the brain, or rapidly organize information. But as for cooler things, like telepathy, or clarvoiance, or any kind of film-worthy mental powers: nope!