Star Trek Into Darkness has been nominated for four Saturn Awards! It's been nominated in the following categories:
Best Science Fiction Film - Star Trek Into Darkness
Best Director - J.J. Abrams
Best Supporting Actor - Benedict Cumberbatch as Khan
Best Costume Design - Michael Kaplan
Not sure is it going to win anything. It's up against Gravity and Alfonso Cuarón in the first two categories; and Tom Hiddleston for Loki in the the third. Can't really speak for the last one. But you never know. We live in uncertain times, as a Cardassian tailor once said.
On a related note, the Blu-Ray releases of seasons three, four and five of Star Trek: The Next Generation are nominated in the 'Best Television Release category. I've heard nothing but great things about those sets, so hopefully they'll bring something home in their category.
Winners will be annouced in June, and a full list of the nominations can be found here.