[Templar]

It's kind of sketchy but from what I can tell ASJ, when he was on the show, bragged about his supposed underground fighting credentials and claimed to have killed a guy. JDF saw through this and won't let it go.
While ASJ was in the wrong to lie he was still a teenager when first cast, and well...teenagers make crap up all the time to seem "cool." So while it was silly it's also not something anyone should hold against him twenty years later.
JDF still does though, badmouthing the guy at every opportunity while constantly challenging ASJ to MMA fights while ASJ was oversees with the army and unable to really respond. So it's really a petty feud from when they were both teenagers/young adults that JDF won't let go of while ASJ has by all appearances moved on.

Again though, that's only my understanding of it.