Private rooms for dining are kind of rare in Seattle. Quinn's has a nice private table upstairs, but more on the casual than fine-dining side. There's Volterra, which is always good. The Dahlia Lounge doesn't mess up. Crush is very nice, quiet and elegant. The mezzanine at Tavolata would be less fancy but lots of fun. RN74—brand new, downtown—appears to have a private room—that'd be the most hottt choice for sure, but you'll spend quite a bit. Oh! Madison Park Conservatory is also new and exciting (and verified really good), and they have a great-looking private room upstairs.
Of course, if you want to REALLY spend: Canlis.
If you don't require your own separate space, you might consider Boat Street or La Bete or the Book Bindery.
More on all these here:
http://www.thestranger.com/seattle/Search?search=restaurants
Felicitations to the happy couple!