I think the fact Scotty specifically said he gave them a good home. Kirk, who had been shocked a minute earlier when he thought Scott had beamed them into pace, and Uhura, who was totally bonkers over the things and she horrified at the though of McCoy dissecting one earlier in the episode were both now laughing. Plus Spock didn't bother to point the logic that the Klingons would probably just kill them all, when he's had no hesitation in stating the unpleasant truth before,....I think that all strongly indicated noone seemed to think a mass killing was about to take place.
Well, in the crew's defense, it's not like we saw the Klingons take a single aggressive action against the tribbles in the episode itself... their main reaction seemed to be recoiling in fear or disgust, and attempting to avoid even the slightest contact. Maybe the Enterprise crew figured the Klingons would abandon ship just to get away from the tribbles, or something.