It wouldn't work, for the simple reason that the styles of the two are so different. Star Trek characters (Starfleet anyway) are rather po-faced, whereas the Doctor comes across as a carefree mischief-maker.
Basically the Trek characters would treat him like Q, Trelane, The Outrageous Okona, or Harry Mudd, and they would be like any number of the various blustering space ship crews the Doctor has met over the years.
I like the idea of the pastiche
of a Star Trek-style ship.