I wouldnt want it across the whole server - because everyone have the right to choose. But I could certainly see them having instances reserved just for a PvP universe - or some kind of option before you start playing. There are ways to get around forcing EVERYONE into open PvP.
They could have home systems which are close to Sol as PvP "safe" and everywhere else could be open pvp. I mean in the show the Romulans/Klingons etc wouldnt turn up and then say oh no..hang on, we arent allowed to attack here. They would attack first and ask questions later. There arent too many Star Trek episodes which contain no fighting/battles/warfare and by and large the most highly rated episodes are the ones which involve exactly that (Best of Both Worlds, the Dominion battle episodes etc etc).
However, I dont think the game will go in that direction, but thats ok, there are after all other titles that can/will provide that kind of focus. As long as STO proves to be a success thats all that really matters. We need more successful MMOs on the market because the MMO market is getting dreadfully stale currently.