I don't think Star Trek lends itself to a detective story plot.  
It has transporters, replicators, error-free lie-detectors, holo-decks, consequence-free time travel, and a few other technologies that i probably forgot.  
The scripts would quickly devolve into the "Deus Ex Machina" of the week.  The mystery-solving episodes of the existing series are some of the weakest.
I think doing a "Civilian" plotline (ala Firefly) would be very interesting in almost any era.  Or several sets of mini-series focused on one group of characters at a time (like american horror story, or the old Outer Limits)