Yes. Knowledge of Q is widespread among Starfleet personnel.
JANEWAY: Look, I don't know what you want here, but I know who you are. Every captain in Starfleet has been briefed about your
appearances on the Enterprise, and I warn you...
QUINN: My appearances? Oh, you've mistaken me for. Oh, well, no matter,
Voy: Death Wish
Apparently these briefings include specific mention of the Q Continuum
JANEWAY: The hell you will. The vaunted Q Continuum. Self-anointed guardians of the universe. How dare you come aboard
this ship and endanger this crew with your personal tug of war.
Those same briefings were also given to Sisko
O'BRIEN: We've got a problem. sir. I just saw Q on the Promenade.
SISKO: Q? Here?
KIRA: What's Q?
SISKO: A powerful and extremely unpredictable entity. I was at a Starfleet briefing on him two years ago.