We have no indication that everything was hunky dory with the Klingons after STV. For all we know, the reception aboard the Enterprise-A at the end of the film ended with a gigantic bar brawl, or there was some heinous diplomatic incident that soured Federation/Klingon relations soon after. All we really need to know is that the Klingons and Federation aren't in a great place diplomatically at the beginning of STVI, and it's not hard to imagine any number of reasons as to how or why that happened.
Star Trek V is set in 2287 according to the official ST Chronology. STVI is set in 2293. That's six years. A LOT can happen in six years.
Honestly, you don't even have to dig that deep. Kirk made nice with a retired and cast out General and a loose canon soldier. There's nothing that says they marched to the high counsel and convinced them the Federation was amazing. Individual soldiers can make friends but governments continue to fight. Besides, Koord was obviously changed by Sybok's meld. Nobody said Klaa was anything other than biting his tongue until he got the hell off the Enterprise.
Kirk and Kang didn't sway anything at the end of Day of the Dove. They just put aside their fighting because they had better fish to fry.