BLUF: The Star Trek Universe and its parallels (e.g. the Mirror Universe) are part of a simulation, and the Q are AI programs that can interact and influence them all.
I started thinking about the nature of the Q the other day when rewatching the Voyager episode "The Q and the Grey"; it seems odd that an omnipotent/omnipresent species would also be so rigid and unimaginative. Why, for instance, do the Q have such strict rules about suicide? You can argue that they fear their species eventually being gone because they all eventually opt to off themselves out of boredom with existence, but as long as they had the ability to die that is unavoidably going to be the eventual outcome of an organism that can exist for infinity (mind-boggling, isn't it?) anyway.
And why is it such a disruptive event in the Q Continuum for two Q to have an offspring? Obviously a species that can live forever doesn't need to procreate, but it's clearly possible so why is it so shocking?
Then you have the fact that the Q are not just invisible observers of the universe; they interact with species and even individuals to, seemingly, drive outcomes that are possibly part of a preferred order to things. Quinn influenced Earth's history to drive certain changes; helping Newton define gravity was certainly a big event in human scientific growth. Why, though? What is driving the Q to pursue certain outcomes in evolutionary history?
It's almost as if the Q have an inherent nature, and are just after an incalculable period of time starting to question and even push back against that nature. So where did that nature come from? Was it possible programmed into them?
Now step away from the Q and look at the characteristics of the Star Trek universe. The Milky Way galaxy alone is chock full of humanoid alien species that (for the most part and with variations/outliers) breath the same air, can live within a certain range of temperatures, have brains/emotions/standards that function similarly enough that ideas and principles can be shared and abided by, and interbreeding is even possible to a certain degree without outside support. That's huge. And though the show tried to explain this a little bit in "The Chase", I think it's fair to say that we've seen humanoid species across the galaxy to a degree that a shared ancestor isn't enough to explain it all.
Beyond biology, let's also look at the makeup of the physical universe: subspace allows interstellar communities to function without succumbing to the laws of relativity: FTL travel is possible; real-time or at least near real-time interstellar communications function; the materials needed to facilitate warp drive are ubiquitous enough that nearly any warp-capable species that wants to travel into space can get it. It is arguably miraculous that the universe in Star Trek can operate the way that it can, and these things do not conform to our current understanding of how the universe and space/time operate.
Taking all of this together, I propose that the Star Trek universe is a complex simulation, designed purposefully to simulate what the universe would be like if it operated by certain principles that would allow for interstellar communities to form and that many humanoid species existed and were capable of interacting with each other. Several instances of this simulation are running simultaneous with similar parameters but modified to generate different outcomes: the Mirror Universe, for instance, has different presets to create more aggressive humanoids less likely to work together. Furthermore, I submit that the Q are AI programs designed to interact with and shepherd these communities into developing in such a way that they would eventually develop the right levels of technology and understanding to grow into a spacefaring species.
I suggest further that the Q are only partially aware of their nature and don't necessarily understand that the universe is a simulation, or if they do they don't share that fact. Over the millennia that the simulation has been running (at least internal to the simulation) the Q individually have started to develop interests and ideas beyond their original programming, and other Q programs cannot abide that because it goes against their designed nature. Imagine a piece of an Operating System deciding it wanted to commit suicide, or two programs getting together to code their own, new piece of software. It's unheard of, and that's why it's so disruptive to the Continuum.
Maybe one of the original designers of the simulation was a fan of old science fiction stories and incorporated ideas from a writer named Benny Russell into the coding of the simulation to recreate some aspects of his work?
I know this isn't a provable hypothesis and it's really just an idea for fun, but it kind of (in my opinion) adds another dimension (no pun intended) to a species like the Q.