Yeah, the only remotely camp moment of Galaxy Quest was Rickman still wearing the Dr Lazerus headpiece while at home.
The clips we saw of the in-universe Galaxy Quest TV series were definitely campy, but that was to set up the cast members' embarrassment and disdain toward the show, an opinion they changed due to the movie's events.
That’s exactly what camp means. It’s about loving something not despite its lack of quality, but because of it.
I suppose it can be done that way, but that's not its intrinsic definition. Camp is something that makes fun of its own characters and situations, particularly when the characters themselves are not in on the joke. But the whole point of Galaxy Quest is to take something we're used to making fun of and show us, and the film's own characters, that it's actually valid and worth believing in. Its endpoint is sincerity, and that's not true of camp.