Is there a relationship between quantum physics and chaos theory on a classical scale? Im a complete physics lay person and I read somewhere that chaotic systems are subject to tiny differences in initial conditions and that the brain is a chaotic system.
Does that mean our thoughts are subject to quantum randomness?
 A: As gandalf61 has pointed out, there are various opinion on this. You can find a very interesting review with many of them here.
As for me, I will try to draft an intuitive argument in terms of chaos. The general idea is that a macroscopic system may be influenced by quantum randomness in some cases when the timescale between the initial conditions and the final state is great enough.
The polish theoretical physicist W.H. Zurek wrote a very interesting article in 1998. There is an example of the Saturn's moon Hyperion being subject to quantum indeterminacy in its rotation state in a very specific sense. Some ideas of this article have been questioned, but I found this review which makes clear the differences between authors. I am not an expert in quantum chaos (which, as you would imagine, is difficult), but I will try to make an argument regarding to your question.
Sensitivity to initial conditions
Every (non-trivial) dynamical system is subject to initial conditions. What is special of chaotic systems is that, according to their laws of motion, the influence of the initial conditions is not constant, but gets greater over time.
Imagine a system which can be in many possible states, some of which are alike, some really different from each other. Through a parametrization of the physical quantitites of relevance, one can describe this states rigorously, inside what is called the phase space of the system. Then, one can define something like a "distance" between different states, measured in this phase space, to account for their physical differences.
Now, imagine you have two copies of a physical system. You put them in very similar initial conditions, i.e. in states which are really close in the phase space. Now you let them evolve deterministically, according to their laws of motion. At first they look very similar, almost identical. But if this kind of system is chaotic, then after some time they will start to get different, further and further in the phase space. This can be described through the Lyapunov exponent
$$ \vert \delta Z(t) \vert \approx e^{\lambda t} \vert \delta Z_0 \vert .$$
Which essentially describes how initial differences between pairs of states can become exponentially greater over time. So, you can imagine that if you wait enough, you can see two "arbitrarily" close initial states to diverge into different states (given they are in a region of the phase space subject to chaotic behavior. Either way, all the evolution would be deterministic.
Quantum "randomness"
The relation to quantum mechanics comes here: Is there a quantum limit on this "arbitrarily" small distance two different states can be apart in the phase space? It is clear that for sufficiently large timescales, our methods of measuring the relevant variables are not precise enough to relate initial conditions to final states. But if the phase space is quantized, then Heisenberg's uncertainty principle would put a lower limit to "what can be measured", or in the Conpenhaguen interpretation, not even "nature" would know what final state will emerge.
To put it the other way: how much time does one have to wait until a difference "smaller" than what can be described according to Heisenberg's uncertainty principle manifests itself greater than that? In the case of Hyperion, this time was estimated (by Zurek) to be of 20 years.
I believe that an argument for "quantum randomness" taking part in our thoughts is related to this. It is evident that there are some states of the brain which are highly deterministic (e.g. when you are swimming) But as it is a really complex system, its associated "Lyapunov time" for some sets of initial conditions may be really short. There are some documents regarding to this, as this article, or the one cited at the beginning.
I think going further (i.e. trying to demonstrate the existence of free will) is speculation, or at least is "highly sensitive" to the interpretation of Quantum Mechanics taken. But I think that it leaves a lot of space for this phenomena: if the a-causal nature of quantum events can have macroscopic implications, this would permit an a-causal connection between the brain and some kind of metaphysical "mind". As I say, there is a lot of discussion which you can know of here. Let me cite some of it to finish:

Planck's constant is an extremely tiny number on the scale of human events [...]. However, QM involves subtle non-local entanglement of physical quantities with possibly macroscopic manifestations [...]. A universal criterion for getting rid of those effects remains elusive [...] and quantum theory should not be easily neglected.

A: We don’t know to what extent (if at all) our thoughts are influenced by quantum phenomena.
Some scientists, including Nobel laureate Roger Penrose, are convinced that quantum phenomena are a fundamental component of consciousness. Others, including Max Tegmark, argue that neurons are too large and too slow to be significantly affected by quantum-level events.
A: I would say that the brain as a physical system is affected by quantum randomness.
But our thoughts are a different thing, thoughts are immaterial, non-physical processes. Mental processes are not governed by causality or any other law of physics.
But I do believe that our minds do use thermal or electromagnetic noise in the brain in the creation of new ideas.
