It's a simple question really. Why do Klingons hate Tribbles, those cuddly balls of fluff from the Star Trek episode "The Trouble with Tribbles"?
Tribbles occur in three episodes in the Star Trek franchise: The Trouble with Tribbles from the original series, More Tribbles, More Trouble from The Animated Series, and again in Trials and Tribble-ations in DS9, which involves time travel back to The Trouble with Tribbles. They're one of the most memorable creatures from the original series, perhaps only outdone by the classic Gorn fight. But where the Gorn is every bit a guy in a rubber suit, tribbles are cute little balls of fluff. Think of it like a cat without the pointy bits, and with a constant desire to cuddle.
Unless you're a Klingon. Tribbles hate Klingons, and Klingons hate tribbles. There have been a few explanations why. Some have suggested that both species have highly advanced senses of smell, and can't stand each other's stench. This seems rather unlikely. Why should a tribble be much different from any other small mammalian style, furry creature?
What is known, is that a tribble infestation affected several Klingon colonies, destroying their crops and their economies. This kicked off the Great Tribble Hunt, where Klingon warriors exterminated all tribbles, causing a vast genocide, and wiping them out completely.
How did this infestation start, exactly? Well, neither Uhura, Scotty, nor Spock really want to take the credit for it, but all the tribbles were transported from the Enterprise to a visiting Klingon cruiser. So, one could argue that the crew of the Enterprise was indirectly responsible for the chain of events which led to the mass genocide of the tribble species. No wonder that none of them wants to take the credit.
But the reason the Klingons began the extermination of thee tribbles was this future infestation, yet there still appears to be some intense dislike between them already. It is quite possible that this was not the first tribble infestation the Klingons have encountered in the past, and that there had been previous, unsuccessful extermination attempts.
Interestingly, in More Tribbles, More Trouble, it is said that the Klingon infestations were due to Cyrano Jones, a human trader, rather than the crew of the Enterprise. This seems to be an attempt to avoid the crew taking responsibility for an ecological disaster, even though they end up sending more tribbles home with the Klingons at the end of the episode.
It seems likely that there have been multiple incidents in the past, which have led the Klingons to label tribbles an ecological menace. The Klingons had already started a genetic engineering project to attempt to eradicate the creatures. It is interesting to note, however, that it is only the Klingons who have such an intense hatred for tribbles.
In the end, I like to think of the Klingon-Tribble relationship like that of an adult with no experience with children, and a baby. Everyone else coos over the baby, which makes happy baby noises, until the adult tries to hold it. At this point, the happy noises change to incessant squalling. Needless to say, both parties develop a dislike for each other.