I dream about whatever I watch on TV, so I have Star Trek dreams all the time.
I wish I could do that!
I've had only two Trek dreams that I can remember. The first was when I was a kid and TOS was the only Star Trek series available. I managed to encounter a Starfleet landing party that was trying to prevent aliens from corrupting the past (my present). I ended up injured by rock debris from a phaser blast, as a bystander cowering in some nearby bushes. They defeated the aliens but realized I was hurt and had to be taken care of. So they beamed me aboard. Dr. McCoy took care of me and then later on Captain Kirk stopped by to check on me. He was genuinely concerned, easy to talk with... And he ended up letting me stay on board to recover while they went on another mission. I got to go through the ship's libraries and learn all kinds of things about the ship and Starfleet technology, similar to how Gary Mitchell did it in sickbay (but of course not nearly as fast). Kirk had ensign Garrovick check up on me periodically and I was able to talk him into letting me accompany them on an away mission. I got a tricorder and a communicator to use, but no phaser. Then somehow I managed to slip and fall down a crevice that formed when the ground gave way. Nobody saw me. The communicator was damaged and could receive but not transmit. And there I sat in a pit, unable to be located because of the composition of the surrounding rock materials. The only other thing I can remember is hearing Captain Kirk on the communicator, apologizing for not being able to find me, but that they had no choice to leave on an urgent assignment. Trapped on an alien world with no hope... So what started out as a fun dream turned into a nightmare.