The “no free will” argument just relies on the brain strictly following the laws of physics.
As for explaining it, I guess it's more of a definitional problem than a physical one. I'm still confident that one day we'll be able to look at something and approximate how conscious it is.
Free will, on the other hand, requires supernatural phenomenons coming from outside the material world (dualism) to explain how we, humans, can make decisions free of any (or some) of the influences of the physical world. I don't buy it, and I believe the burden of proof falls on the ones who do.
I kind of do believe in a deterministic universe, but I thought quantum physics disagree with that.
But I think this is irrelevant to the question of free will: having quantum states randomly collapse insinde my brain would have nothing to do with freedom.
Not necessarily.
Most people learn the "Copenhagen interpretation" in intro quantum mechanics. The quantum wave function then evolves deterministically (Schrodinger equation), but collapses probabilistically (on observation). In that sense, the world is fundamentally random – although I personally wouldn't describe a robot driven by a real RNG as having "free will", even though it is indeed non-deterministic.
However, there is a major caveat here: This is just one interpretation of quantum mechanics, and there are fully deterministic alternatives. For instance, in "non-local hidden variable" and "superdeterministic" interpretations, the wave function collapse is believed to be a deterministic process. In the "many worlds" interpretation, the wave function doesn't collapse at all (it only entangles), and you're left with only the deterministic Schrodinger equation. It has also been shown that quantum-like behavior can arise in deterministic systems, see e.g. the "bouncing droplet" experiments on YouTube if you're interested (which is a beautiful macroscopic analogue to the "pilot wave" interpretation, which is a viable "non-local hidden variable" theory).
The reason it's commonly stated that quantum mechanics requires non-determinism is that (i) the philosophically most appealing "local hidden variable" theory has been falsified, (ii) the Copenhagen interpretation is easy to teach, and (iii) for practical calculations it actually doesn't matter.
(Source: I work in quantum mechanics.)