One studio (CBS) owns the rights to the televised Star Trek universe, and another (Paramount) to the movie Star Trek universe. But to make it even more complicated, the movie rights trump the TV rights, so elements that were in both (such as TOS and TNG) can't be in any new series without Paramount's permission.
Both studios used to be owned by the same company, but they split a few years ago, around the time Enterprise was ending its run on TV. A rumored merger of CBS and Paramount vis-a-vi the Viacom merger would have reunited the rights, but it looks like that is permanently off the table (see the Redstone litigation).