The answer to this question is generally not a simple one.
The specific Q entity portrayed by 'John de Lancie' has made references to the true nature of the Q and their existence starting from his first appearance in the TNG episode "Encounter at FarPoint"; that the human mind is not capable of comprehending the true nature of much of the universe (the crew's confusion about the nature of the events supports the theory).
In fact, every time Q appears something happens that is well beyond human comprehension and in one occasion the TNG crew find themselves in Sherwood Forest.
In the episode "Deja Q" de Lancie's character has been stripped of most of his abilities by the other Q and is struggling to understand what the beings he has been "tormenting" are actually capable of. He says to "change the gravitational constant of the Universe" as a solution to a problem because altering the universe, and all of the subspace and technology within it, is something that would otherwise have been trivial for a Q. Changing the universal constant, however, is well beyond anything Star Fleet and their understanding subspace can achieve so they attempt to change the moon's gravitational constant instead (Note that the Federation often alters the mass, inertial and gravitational characteristics by pushing the ship and its contents into a subspace state).
In VOY episodes such as "The Q and the Grey", the Voyager crew end up visiting the continuum and it is explained that the reason the continuum seems to be a re-enactment of the American Civil War (with Voyager nowhere to be seen) is that their human minds are not capable of understanding what is really going on so it merely appears in simplified terms (usually something out of Earth history). By contrast subspace is depicted as a region through which space ships and other objects can travel (not wooded or desolate land in the limited human understanding). Further it is also referenced that the Civil War style weaponry that everyone is firing at each other are actually disrupting normal space and causing supernovae all over the place that are actually apertures into the continuum and that Voyager can only enter by flying through the Supernova that occurred in the aftermath (likely what the shield was needed for). I believe Q makes a reference that they should go nowhere near the actual business end of the weapons considering what it would do to a human if it can actually make him seem to "bleed".
What the writers seem to be getting at here is an that there are a very limited number of effects that can actually be visually produced by any character (either in universe or by the show's special effects crew); whatever coincidence there are in appearances the Q are NOT using any method or concept known to the Federation nor is the continuum anywhere that the Federation would be able to name. The Q ARE however often seen appearing on ships within subspace (Deja Q, The Q and the Grey and several others) and capable of using subspace techniques and technologies to do what they desire should they so choose.
