Uhm Viacom/Paramount OWNS Star Trek
It's actually CBS that owns Star Trek... just sayin.
They both do.
Paramount owns the JJ Abram's version of the current feature film Star Trek franchise.
CBS owns all the previous versions of Star Trek (all 6 TV series: TOS, TAS, TNG, DS9, VOY, ENT and the TOS based and TNG based feature films.)
That said, BOTH companies have worked together to cross promote where possible and help maximize exposure and profit for one another across the board. I also think CBS is waiting to do anything new (production wise) with Star Trek on TV, until the current series of Paramount Star Trek films have run their full course. Paramount and CBS could always strike a deal and produce a new TV series more in line with the JJ Abrams film universe; or CBS could bring something out with ties to the 'original' universe - or they could just continue to remarket the 700+ hours of TV episodes and the TOS/TNG films going forward.
Time will tell.