I would propose Qs are capable of doing this, but its not something they actively engage in. I'd suggest that their arrogance leads them to believe there's no practical reason (or mischievous) reason for doing it that holds their interest.
Do keep in mind, the Q lack some very basic understandings since they came into existence with the power they have. They never learned lessons like "Ever vigilance" or how it can be helpful and even essential to monitor and gather information. I would think they would, again, write it off as a waste of time because if they really want to know something they can go back or forward in time to find out, but "Databasing" the knowledge isn't something they see as practical. Its kind of like the same effect most people have with wikipedia and similar sources today. Many (not all) people don't bother committing things to memory because they can look up the information whenever they want, but, again, many people instead of accessing the information when they actually need it, simply don't, choosing to remain ignorant (more then likely because they feel rushed for time for other reasons).