Does a quantum computer resolve the halting problem and would that advance strong AI? Have there been proposed extensions to go beyond a Turing machine that solve the halting problem and if so, would those proposed extensions have value to advance strong Artificial Intelligence?  For example, does quantum computing go beyond the definition of a Turing machine and resolve the halting problem, and does that help in creating strong AI?
It depends a bit on what you mean by 'quantum computer'. The 'conventional' notion is that quantum computation buys a (in some cases, exponential) speedup - it doesn't change what can be computed, just how quickly.

In contrast, advocates of hypercomputation claim that quantum effects may make it possible to do infinite computations in finite time. Note, however, that this is not a mainstream belief - the reknowned logician Martin Davis has written an article claiming that hypercomputation is a myth.

Roger Penrose has also claimed that quantum vibrations in neural microtubules may be responsible for consciousness.