Consciousness is often used to describe Śūnyatā by some teachers but in Buddhism consciousness is one of the aggregates.
The way these relate, according to instructions I received, and my limited understanding:
In ontological reality nothing exists by itself; everything is part of some process and relationship; In fact in Mahayana Buddhism we say that delineating these processes and relationships into distinct entities is a function of the mind. Same processes and relationships can be delineated in a multitude of ways, even though most of the time we assume there is only one, or only one right way to delineate them. Entities do not have (are empty of) inherent existence and are delineations made by the mind. We call this situation "Emptiness" (Śūnyatā).
The mind itself exists as (informational) phenomenon emerging from interaction of processes and relationships. So mind itself is not something fixed that has essence - it's fluid and ephemeral. We say, mind itself is empty.
Since "entities" do not exist as anything apart from the mind, and mind itself does not exist apart from the interaction of processes and relationships, we can see that mind and entities are of the same emergent nature. In other words, the latent regularities inherent in the interaction of processes and relationships give rise to mind as reflection and representation, culminating in delineation of entities.
So mind and its delineated entities emerge as virtual aspect of regularities inherent to the ontological reality. Mind and entities are virtual. This virtuality is what we call Emptiness. In this sense we can roughly say that mind is emptiness and emptiness is mind. Or, more precisely, that the nature of mind is emptiness. So in the absolute sense they are inseparable.
However, when we speak in the conventional sense, mind refers to representation, and when we speak about vijnana we specifically mean representation that develops in the case of sentient beings. The capacity for representation is one of the five components making up the subjective realm of the so-called sentient being, along with stimuli, imprints/tendencies, concepts of entities, and feelings. Vijnana is that which emerges from imprints/tendencies in response to stimuli and underlies the process of delineation and evaluation of entities, as well as the feelings that fall out of that. So in conventional use the meaning of "Vijnana" is rather specific.
While Emptiness (Śūnyatā) is a much more broad concept. It refers to the global situation that ontological reality does not have any sort of absolute point of reference, whether local or global, and that whatever virtual stuff that emerges, emerges from relationships and interactions, and that therefore all phenomena, including the ones we deal with in our human realms do not have any sort of inherent solidity.
They are very different terms with very different meanings.
Mind (pali: viññāṇa) is that which knows something. Emptiness (pali: suññatā) is how things exist. We say the mind is empty - ie., that it lacks inherent existence. Madhyamakas say that all things lack inherent existence... ie., that they are empty of inherent existence. The question then becomes what is 'inherent existence' ;)