I second PG, it depends on your budget. There are plenty of elegant restaurants. But if you want to have a glass of wine with your meal, it only can be the one at some 5 stars hotel.
"Fish Market" in Katara: food is good, but i wouldn't call it "elegant" under any circumstances. It's rather a real "market" for big loud families.. But again, food is excellent! No alcohol though.
One of the elegant restaurants is Maison du Caviar in W. For instance, grilled salmon is about QAR.140, salads are around QAR.50-70, and so on. Great selection of wines.. If it suits your budget, that's a great place for quite romantic evening for two of you.
Other choices are:
-"360" in Torch (prices are about the same, no alcohol though);
-"Al Moorjan" on the Corniche, with beautiful seaview(sample: grilled shrimps jambo is QAR.180, lobster is QAR.320, etc.) - no alcohol
- "New York Steakhouse" at the Renaissance hotel.. excellent steaks, as well as great seafood..
I can continue, but you have to specify your budget and preferences..