It doesn't seem likely. There are practical reasons why the shows were ended after seven years. Actors and other staffers get raises every year they stay with the show, so unless a show rotates its cast pretty often, it will eventually get prohibitively expensive to keep making it. (Smallville ran for 10 years, but largely because it only had one of its original cast members left as a regular by its final season.) And just about any show's ratings will fall over time, so generally they have to cut the budget more and more with each successive season in order to survive.
So the formula for an extended lifespan for a Trek series would entail 1) getting rid of a lot of the actors and replacing them with new, cheaper ones and 2) slashing the budget and reducing the amount of special effects, location work, action, elaborate sets, guest stars, etc. they can afford to fit into each episode. The question is, would that be desirable? I'd say no. Better to wrap up the show while it's still affordable to make it impressive.
I wouldn't mind that, actually...
Periodically replace the actors as the show continues. Face it, people get transferred, new crew members come on board. it would be highly realistic to see that Captain Johnson gets promoted to admiral, Commander Jones gets promoted to Captain, new first officer comes on, and you can still have the now Admiral Johnson appear every now and then as a guest.
Hell, even Next Gen came close to doing in the Best of Both Worlds.