Does 'the quantum mind' solve the hard problem of consciousness ?
No it doesn't.
Penrose's quantum mind model is an answer to the question of how can a human mind perform computations that a computer (more specifically a Turing machine) can't. This is known as the Lucas' argument against mechanism. There are several criticism's of Penrose's theory, especially since it is not established that the human can do anything that a computer can't. See this reply for details.
More generally, quantum theories of mind all share a common feature: That the mind has features that can't be explained by classical physical models, but can be explained by quantum physical models, presumably in the same way that electron interference patterns or the tunnel effect cannot be explained by classical models, but can be explained by quantum models.
Such models still fall in the domain of physics, they just require quantum physics.
On the other hand, the hard problem of consciousness centers around the fact that consciousness (qualia, subjective first person expereince, etc..) for the time being can't be explained by any physics, not classical, not quantum, not relativistic or anything.
David Chalmers, who formulated the modern version of the problem, states "In Facing Up to the Problem of Consciousness (1995)":
It is undeniable that some organisms are subjects of experience. But the question of how it is that these systems are subjects of experience is perplexing. Why is it that when our cognitive systems engage in visual and auditory information-processing, we have visual or auditory experience: the quality of deep blue, the sensation of middle C? How can we explain why there is something it is like to entertain a mental image, or to experience an emotion? It is widely agreed that experience arises from a physical basis, but we have no good explanation of why and how it so arises. Why should physical processing give rise to a rich inner life at all? It seems objectively unreasonable that it should, and yet it does.
Hence the idea of philosophical zombies: A purely physical description of the mind fails to distinguish between a conscious person and a zombie who only seems conscious but has no qualia or inner experience.
Another way of illustrating this is Frank Jackson's knowledge argument:
Mary is a brilliant neuroscientist who knows all the physical facts (inclduing classical, quantum, relativistic, and any other possible theories) about the color red, as well as all of the related biological and neurological facts. But she herself has been color blind since birth. Because of this, there is something missing from her knowledge of the color red, namely the qualia of red, even though she knows all the physical aspects of it.
and how popular is it?
Quantum theories of mind seem to get a lot of mention in popular literature, but don't really get that much traction among academic philosophers of mind. Penrose himself is a mathematician and physicist, not a philosopher, and his take on Lucas's Godelian argument is not agreed on by most theoretical computer scientists either. In particular the consensus is that quantum Turing machines are equivalent to in power to classical Turing machines, so adding quantum effects to the mind won't make it in any way more powerful than a computer.
Other quantum theories of mind, similarly don't seem to get much attention.
That being said, here's an interesting dissenting view.