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I don't know the specifics of your story, or how "close" it is to Star Trek, but from the way you describe it, I would be almost positive that you need permission, or you would be running into some major copyright problems. You could be running into problems with character copyrights, story-lines---any number of different things protected by copyright. On the other hand, you may be able to obtain permission to use some of the copyrighted info, quite easily.
I wish you the best of luck on your venture.