I encourage you to create your own universe. Many fan films have done that, and it makes you far less likely to get on the wrong side of the owners of Trek.
Here are two which were adopted by more than one fan filmmaker:
Star Track http://startrekreviewed.blogspot.com/2009/06/96.html
. Note that this fan film series had enough devoted fans than another group created their own series in the Star Track universe: http://startrekreviewed.blogspot.com/2009/06/97.html
Star Trekz Empire: http://startrekreviewed.blogspot.com/2009/06/124.html
and the related (possibly same universe, but different producer and writer) Fem Trekz: http://startrekreviewed.blogspot.com/2009/06/117.html