Theres a quote that stuck in my head for a while,
It says something about Garen coming back to Demacia tired after fighting Kat, but he's never really been tired before, according to the lore. It says ''he came back tired, but his fellow soldiers suspected different reasons'' or something along those lines.
Kat x Garen is canon?
and 
might have given us hints based on their taunt quotes towards these two. And long time ago, back then when there use to be this thing called "Summoner's Issue," 
has a dating service that showed the two actually being a couple. In another "Summoner's Issue," a few guards were about to attack Kat until Garen stopped them. Either it's something personal or he's just trying to keep the uneasy truce between Demacia and Noxus. They did went to war once in Kathamandu which was eventually changed into the Crystal Scar. There's more I'm probably not aware. Better keep researching to see more about the lore.
It's pretty much confirmed at this point that Garen and Katarina are in love with each other,but we don't know if they actually had sex(or if they ever met outside of the battlefield at all)
