If it's your own art, not copied or re-drawn but a true original, then yes. It's how sites like TeeFury are able to operate. I wouldn't flat-out call it "Enterprise" or mention anything specifically Star Trekky, though.
They are not "Allowed" anything if they did not get consent from the copyright holder. The fact that they are still around just means that they are not worth enough money to pursue, or they dont want bad publicity smashing the crap out of someone doing "fan art".
If the copyright holders wanted to shut down TeeFury with a DMCA notice, there would be very little problem showing the infringement.
Long story short, if you make money by using/copying other people artwork, you are liable for a lawsuit, they can take every cent you made in some cases.
If you are sitting on Millions of dollars, and a crack lawyer team working for you then I would say, yeah! give it a shot! You can probably beat them in court!
If you are just a small time guy, making an indie studio though, you might spend your time working on your own artwork and concepts. That way no one can take your hard work away from you. Even if you make $0 using the content, they can still file a DMCA take-down notice.
Stealing artwork and other peoples ideas takes all the sport out of making video games anyway.