In the Star Trek universe, the Mirror Universe is depicted as a parallel reality with significant differences from the main Star Trek universe. While the Q Continuum and the Q themselves have not been specifically featured in the Mirror Universe on screen or in canonical materials, it's difficult to definitively say whether they would exist or not. The Mirror Universe is known for having alternate versions of many characters and species, but there is no direct evidence regarding the presence of the Q.
However, considering the omnipotent and extradimensional nature of the Q, it is reasonable to speculate that they might exist across multiple universes, including the Mirror Universe. The Q are often portrayed as having knowledge and abilities that transcend traditional concepts of time and space, so they could potentially be aware of other universes and their counterparts within them.
It's important to note that Star Trek canon is extensive and includes various series, movies, and novels. While the Q and the Mirror Universe have been explored in different Star Trek media, the specific interaction or awareness between them has not been explicitly established. As a result, any conclusions about their relationship or existence in each other's realms would be speculative.