Given the Tribbles are an analogy of the 'Rabbit problem', it's not unreasonable to expect that the Klingons would do exactly what humans did: take any measure necessary to cull the ecological menace.
I agree that Scotty beaming them over to the Klingon ship, lighthearted though its played, should have started a major diplomatic incident. It'd be like if Australia had just sent all the rabbits to Indonesia and said "Not our problem anymore, deal with it yourselves". There'd be hell to pay.
Well given the fact that a Klingon plot to kill, at the least, tens of thousands of people with poisoned wheat had just been foiled, I don't think the Klingons would really have a leg to stand on diplomatically if they got pissed and wanted to make Scotty giving them the Tribbles into some big issue.
And I really don't have a problem with the Klingons killing the little buggers.....My issue is noone on the Enterprise, who all seemed to be concerned for the welfare of the Tribbles of varying degrees, seems to have even had the slightest thought cross their mind that the Klingons were going to wipe them out and they all acted like this was some wonderful solution that would be great for the furballs. Like I said not even Spock, who never hesitated to point out the unpleasant, seemed to grasp the logic of what the outcome would be.