The scene when Tim Allen is in a men's room overhearing how the cast of Galaxy Quest are nobodies and all the co-stars can't stand him mirrors an actual event in William Shatner's life. He discovered the exact same things about himself when he attended a 1986 convention.
Tim Allen admitted that he was quite star-struck when he met Sigourney Weaver, as he's a huge fan of Alien (1979). Allen even got Weaver to sign some of his Alien memorabilia between takes. She ultimately did, writing "Stolen by Tim Allen; Love, Sigourney Weaver", which she said made him very upset.
According to director Dean Parisot, Tim Allen became very emotional after filming the scene where Jason reveals the truth to Malthasar, saying, "Yeah, I don't like these feelings I'm having. I'd like to go back to the trailer." After he left, Alan Rickman quipped, "Oh my God, I think he just experienced acting." Coincidentally, this was the day DreamWorks head Steven Spielberg visited the set. He complimented Allen's acting.
William Shatner has seen this film and apparently enjoyed the film. He has said how jealous he was that this production was allowed to introduce rock monsters ten years after Paramount Studios refused to let him include them in Star Trek V: The Final Frontier (1989) on account of budgetary issues. He also jokingly said that he had "no idea who Tim Allen was supposed to be portraying but he did recognize someone playing the Nichelle Nichols character."
This film was one of the earliest to have its own domain name and website, GalaxyQuest.com (now available only via Wayback archive). However, rather than being a polished part of the film's marketing campaign, the site (in keeping with the film's fan theme) was deliberately designed to look like a fan page, with screen captures and poor HTML coding.