No. First and foremost, there are no Star Trek shows or movies in production right now.

During production, due to the nature of shooting, insurance, and the distraction to the actors, there are no public tours available of the Star Trek sets. Space is tight on the sets, with room for cast and crew, but not much else! Some are complete sets (usually four walls and a ceiling), which would make it difficult to see the action while they are filming anyway — you'd need to be right there with the crew.

Unlike filming for a live audience sitcom, the hour-long Star Trek episodes are shot slowly over seven production days. The filming can seem slow, compared to a quick-paced, audience friendly sitcom like Frasier. These half-hour shows generally take only 2-3 hours to shoot, but they have been rehearsed for the week beforehand.

When filming on a Star Trek show is finished for the season, the sets usually undergo refurbishment to prepare them for the next season. When the show is finished for good, the sets are permanently removed to make space for the next show to use the soundstage. The old sets are either destroyed or put into storage for possible future use.