Seriously, I am not sure if these even qualify as 'jokes' or not. But read on because I know you can't not read them now :)
Question: Where do Star Trek fans go to lift weights?
Answer: The "He's dead, Gym"!
Question: How many ears does Captain Kirk have?
Answer: Three. A right ear. A left ear. And a final front ear.
Question: Did you hear about the singing contest for young men at Starfleet Academy?
Answer: It's called the Kirk Tenor Prize.
Question: How many landing party members does it take to screw in a light bulb?
Answer: Only one, but the extra red-shirt will die in the attempt.
And just for fun, some interesting Star Trek trivia...
In 2000, the show was listed in the Guinness Book of Records as having the largest number of spin-off productions, including the feature film series and the numerous TV series.
Contrary to popular belief, Captain Kirk never said, "Beam me up, Scotty," in any episode
In the hallways of the Enterprise there are tubes marked "GNDN", these initials stand for "goes nowhere does nothing".
When NBC was promoting "Star Trek" (1966) in magazines, all shots of Spock's pointed eyebrows and ears where airbrushed out of the pictures because NBC thought that no one would watch the show due to Spock's resemblance to the Devil.