I’m currently running The 13th Fleet for my home group. It’s a FitD hack that draws heavily on Trek tropes and elements, with each player as the captain of a starship in the eponymous 13th Fleet. The 13th is essentially a dumping ground for problem officers. So all the captains are bastards with dark secrets, or buffoons with overconfidence, and so on.
The game begins with the 13th Fleet as the only surviving fleet from the “Nearly Victorious Battle”. They must flee through occupied space back to federation space. With the rest of the armada missing or destroyed, the captains realize that the position of admiral is up for grabs. So the game requires that the captains work together to navigate the obstacles that face them, but also they’re jockeying for position to be named admiral upon returning home. Part of the game involves them stabbing each other in the back to try and come out on top.
It’s a very tongue-in-cheek take on Trek. Each player is a captain with their own ship, and their own Redshirt minions. Play is broken up into a few distinct phases as they use stargates to jump from system to system. We’ve got a couple of sessions under our belt at this point and I have to say it’s a lot of fun.
If you’re looking for a more satirical take on Trek then this may be a solid option. If it’s too over the top, or you don’t like the PvP elements, I think it could be altered a bit to suit a less absurd approach to Trek.
Otherwise, I’d agree with the recommendation of Lasers and Feelings.