Wow, people are really starting to poke around the edges of what's legal here. I just had a conversation with our entertainment law attorney at work the other day (he cleared all the Namco characters for Wreck-It Ralph, so he knows what he's taking about) and he explained to me the legal realities of some of the stuff that fans do thinking it's okay but which would mean lawsuit if Paramount ever got the slightest itch to do so. He said typically the only reason most companies don't crack down on these things more often is because they fear bad publicity more than such an infringement.
Slight nit-pic: it's CBS that owns "Star Trek" for copyright purposes. (Paramount has a license to produce films)
ANYWAY - I suspect this very thing is why the contest wasn't on their FB page or well advertised. It would have made it too easy for CBS legal to see it.
And, from the mouth of CBS: the only reason they "don't crack down on most of these things" is because they don't get enough publicity, or a large enough audience, for it to matter to them. They do watch - and take action when they think it's big enough to take some spotlight wattage away from JJ, or it's a huge violation of rights, or a violation that is noticeable enough to them to care.