Thomas Shey
Legend
No, they don't. They do things like internal trials for breaches of the Police Services Act (name specific to Canada). A rose by any other name smells like compost.
At least in the U.S.--and TOS was very U.S. centric in its basic assumptions--they still have to be tried by a civilian court after that; all inquests do is decide, at best, if you can get kicked off the force. That pretty clearly wasn't the case there.
Basically, at some point them "not being military" is just an informed trait; it doesn't seem to match their organization, equipment, or methodology under threat in any meaningful way. Its not like some of what they do that isn't strictly military hasn't frequently been one of the military's side jobs, too (diplomatic escort and contact for example). The exploration end hasn't been a thing in the real world for some centuries now, but even when it was, the people doing it might as well have been private military contractors for all intents and purposes (and in few cases, absolutely were). So we're left with the science end being unprecedented for a military, and that's just not enough for me to buy that's not, in practice, what they are. In the TOS era, they weren't even that much slow off the trigger than militaries with strong rules of engagement.