New Star Trek headcanon: since tribbles infest grain supplies in a manner akin to common earth mice or rats, they should thus be ecologically similar - that is to say, cats should perceive tribbles as prey.
Now consider this: when Klingons learn that cats can and do hunt tribbles, they decide to make use of that intel. This leads to a trade deal between the federation and the Klingons, where the latter asks for large numbers of housecats. The federation is confused, but decides to go along with it - what's the worst that could happen, after all? If nothing else, it is an opportunity for trade!
So, the cats are thus recruited into the Klingon war on tribbles. After a hard-fought victory, everyone in the empire celebrates, with the cats being guests of honor more often than not. The Klingons are happy; the cats are happy and fat with tribble meat; the federation looks on in confusion at what just happened.
Thus, the road to the Khitomer Accords has become possible, through the power of housecats - and careful diplomacy, of course. But mostly the cats. Because tribble hunting.