Yeah, as of what has aired anywhere, we know literally nothing about his family. Nada. Zip. Zilch. In his room there's a picture of Joe Flanigan as a kid with Evil Kneivel, so I usually assume that's kid!John and Kneivel. In "Letters from Pegasus," he says something along the lines of "Not everybody's got someone at home," when asked about his message.

Also, the website used to say (may still say, I don't know) that his father was, quote, "a Cold War general," but Joe Mallozzi has explicitly stated that that will be contradicted in an upcoming episode in Season 4, called "Outcast." (there are spoilers at the link)

Meaning that, until later this season, anything you write? Won't contradict canon. :)