Nature of the Q continuum
The Q continuum is an extradimensional realm, and as such exists outside the space-time continuum of the universe as we perceive it.
It is highly likely, as extradimensional beings, that the Q can travel through time in our universe. Our universe is either a spacetime submanifold of the larger Q continuum or are two disconnected universes. In either case, the Q seem to be able to enter our spacetime manifold at a point of their choosing ("point" meaning a location in space and time).
Memory Alpha has the same opinion:
The Q Continuum was an extradimensional plane of existence inhabited by a race of beings known as the Q. The term could also refer to the Q society itself.
As a race, the Q were immortal, but not absolutely omniscient or omnipotent, just possessing the ability of instantaneous matter-energy transformation and teleportation, as well as the ability of time travel. Their apparent abilities included moving entire asteroid belts and stars, creating alternate timelines, and affecting universal states of nature such as the gravitational constant.
Note that these are just my opinions and the opinions of whoever contributed to the Memory Alpha entry.
Knowledge of time travel techniques
Since the Q are for all intents and purposes omniscient and know exactly what humans have and have not achieved, there is no denying that the Q would know that humans themselves have achieved time travel in the past (a feat performed twice by the original Enterprise crew). Therefore, the Q would know how to achieve at least one form of time travel.
Since the question was "Do we really know whether the Q can time-travel?" (rather than, "Have we seen an actual instance of Q travelling through time?"), the answer would almost certainly be Yes.
Possible canon example
In terms of canon, we see in the series finale of TNG ("All Good Things...") an example of Q travelling through time, at least seemingly. He and Picard travel between four different time periods: the present; seven years prior; twenty-five years into the future; and far into the past to a time when life was only beginning on Earth:
On the face of it, there is no indication that these travels were an "illusion". On the other hand, since only Picard remembers what happened, it could have been an elaborate illusion. The sentiments of the official StarTrek.com entry on the episode suggest that it was "real", however:
Picard is returned to the moment his adventure began, coming from the Turbolift in his bathrobe. He shares his experience with the crew, using what he has learned about the future to forge a new, closer relationship with his dearest friends.