Godel's incompleteness theorems apply to pure mathematics, so even without having any idea how our brains work, we still know for sure that they apply to our brains as well. Even if our brains were able to solve the halting problem, they would still not escape the Incompleteness theorems.

And note that it is not that hard, at least in principle, to construct a system that can detect its own incompleteness.

As far as we know today, the most likely theory is that our brains are Turing-equivalent computers. There is no other known model of computation, and no scientific argument that would show our vrains/thought processes can't be computers (that is not to say we have proven either of these, they are just the most likely statements, like P!=NP).