The Borg were never at war with the Klingon Empire in Star Trek, which feels like a missed opportunity for the franchise. While spin-off fiction like the non-canonical novel series Star Trek: Destiny have depicted all-out war between the Borg Collective and their enemies in the Federation and the Klingon Empire, it's never been seen on-screen. As the main antagonists of Star Trek: The Original Series, it would have been interesting to see the Klingons step up to take on the Federation's greatest enemy during the Star Trek: The Next Generation era.
The only notable Klingon to fight the Collective was Lieutenant Worf (Michael Dorn), whose service aboard the USS Enterprise-D meant that he regularly battled the Borg. Apart from Worf, it's unclear exactly how much interaction there was between the Klingon Empire and the Borg Collective. Although they sent ships to assist their Federation allies during the devastating Battle of Wolf 359, they didn't appear on-screen during the fight. Star Trek: Picard season 1 revealed the Romulan interest in the Borg Collective, depicting their experiments aboard the Borg Cube Artifact. However, to date, it's unclear if the Klingons ever launched a full-scale offensive against the Borg Collective.