Finally got time to test it out...
AND IT WORKS!
Ubuntu 13.04, 64bit
AMD Phenom II x4 840
ATI XFX HD6450
2x4GB Corsair DDR3
Steady 30+ frames on lowest settings, but that just means that I have to hurry with GPU upgrade
It was quite easy to set it up as well, Wine 1.7.14 + IE8 + vcrun2008
Only thing that it took quite time to pass loading screen (one with Romulan insignia, about 5 minutes) and had to log in game after it loaded.
About Mac version, I haven't tested it (don't have Mac), but from what I've heard, it's run by Mac version of program similar to Wine?
So it's not fully Mac client and I would not have high hopes for Linux version...
EDIT:
One trick that I've found on
STO WineHQ, and it helps in reducing loading time.
"However, I changed the IE boot from
Native then built in

Built in then native."
It's under Wine Configuration/Libraries/iexplorer.exe and just editing it there works perfectly.