Have you let the Force Verify complete when you relaunched the client?
If yes, you could also try deleting your texture.hogg files. (There are three, and are located in the Live\piggs folders. Sample path: C:\Program Files\Cryptic Studios\Star Trek Online\Live\piggs)
I find that running the game in "Fullscreen Windowed" as found on the top of the graphics setting makes a world of difference for this. It may take a few restarts to get it to stick correctly, but once it does, I have had no issue tabbing out or using programs on a secondary monitor.
I got also a lot of times game crash when i switch to desktop with ALT+TAB at different gameplay moments, eg. i set to fly my ship to some location and ALT+TAB to do some other things (eg. to check STO forum) and 90% of times i unable to back to game, game does show just black screen, play music and feeling like game working but just i see blank, black screen and if to wait a bit nothing happens..
So normally i kill it with task manager and run again... I think game has some problems with it and not tested in this way.