The first, most important factor to address is that the Q are an incredibly powerful, god like race of being, and the likelihood that they could be captured or defeated by the Borg is incredibly unlikely. These omnipotent beings have the power to change the fabric of the universe with just the click of their fingers, having created various timelines over the years just for fun, and literally Thanos-snapped beings out of existence. The Borg, while still a powerful threat, are nothing compared to this type of power. The Q are not only able to defeat their entire race without breaking a sweat, but can also eradicate any trace of their existence altogether.
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The other issue is that there would be nothing for the Borg to assimilate in the first place. The Q’s whom audiences see over the various programs are simply just physical projections of their true selves. They are non-corporal, and simply send avatars of themselves to the universe perceivable to humans and the like, more like a projection than their real consciousness. If the Borg were somehow able to capture one of these corporal projections, and tried to assimilate it, they would just end up with whatever the Q wanted them to have — ranging from a pile of newspaper to nothing at all.
Each and every time a physical being experiences a Q, they are seeing a manifestation the Q has created that the person would understand, their true form being incomprehensible to them. This is shown not only though the Q taking human form for the human captains (even Captain Sisko, a man Q might even be a bit afraid of) but also in the way the Q project their home, the Q continuum, to corporal visitors. In the ill-received Voyager series, their home is shown to be an old run down farm town, something the crew could not only understand, but could physically see and interact with. But of course, this is not its true form.
Ignoring all these incompatibilities between the Q, who are not made of flesh and blood, and the Borg, who consume and assimilate flesh and blood, an interesting question arises. The Borg’s main mission in their destructive sweep across the galaxy is to assimilate in the attempt to devour information, driven forward by the idea of achieving perfection for not only themselves, but for all life in the galaxy. They were, before the introduction of the Borg queen, a wonderfully designed villain that acted more like a rampant computer function, being taken with one simple goal (perfection) and then simply carrying out this function without being clouded by morality. They were not good or evil. Assimilating a Q, then, would potentially fulfill this goal, the Q being the most powerful beings in Star Trek. The Borg would become an unstoppable force, able to assimilate or simply change the galaxy to fulfill their vision of perfection.
In addition to gaining all the knowledge of a particular Q, having their consciousness uploaded to the Borg collective, they would also potentially gain the god-like power of a Q. While having just one Q-powered being as part of the ever-changing Borg armada, it has been shown in canon that the Q are able to take away the powers of another Q. While the knowledge would remain in the collective, their power would theoretically be nullified, and they would be unable to harness it for their desires. The problem, however, is that in addition to the Q being able to take away Q power, they are also able to give it, as was shown in the TNG episode “Hide and Q” where Riker was given their almighty power. The freshly assimilated Q could then, if no other Q intervened, give the Q power to all the Borg drones in the collective, and theoretically create an army of Q Borg so large they could defeat the Q Continuum.
This argument is, of course, hugely theoretical, as the strange series of events that would have to take place in order for a Q to be assimilated is bordering on the ridiculous, even for Star Trek. While it might not even be possible, the only way it might happen is if the Q become so bored they decide to let it happen, or even make it happen, just to see the consequences. It might well have already happened, in one of the various different Q created timelines, but thankfully the prime timeline remains lacking in the almighty nefarious power of the Borg Q.
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