Well, the only place on property that I'm sure you won't find many, if any, children at and is super fancy is Victoria and Albert's in the Grand Floridian, but that's like $100-200 a person.
https://disneyworld.disney.go.com/dining/grand-floridian-resort-and-spa/victoria-and-alberts/ Otherwise California Grille for maybe a late dinner and view of the fireworks (might have to make reservations early because that slot probably fills up fast - although I think I read if you have an earlier dinner time, you can keep your receipt and come back to sit in the lounge to view the fireworks or go on the balcony/viewing platform).
https://disneyworld.disney.go.com/dining/contemporary-resort/california-grill/ Artist Point at the Wilderness Lodge is also a fancy one with a high price range, so you might not find many children there.
https://disneyworld.disney.go.com/dining/wilderness-lodge-resort/artist-point/ Jiko looks really nice, and with the unique menu I don't think many children's palates will like it, so you might not find many there either especially for dinner.
https://disneyworld.disney.go.com/dining/animal-kingdom-lodge/jiko-the-cooking-place/ You can take a look at all of the fine dining places at this link
https://disneyworld.disney.go.com/dining/#/fine-signature-dining/ Just to note, I have never eaten at any of these places as they are out of my price range

That's not to say I wouldn't like to though. I'm just thinking with the high prices, parents wouldn't want to spend that kind of money on bringing children, and I would think most would assume people eating dinner at these places would be doing it for a special or romantic reason rather than a family experience. And, the atmospheres would be more quiet and romantic than a regular-priced restaurant. As far as quiet resorts, I think two of the most quiet are Old Key West and Saratoga Springs. They both have villas which are great for relaxing. The one bedrooms usually have a living room area, kitchen, and a huge bathtub. And I know Old Key West's has balconies. If you're into rustic, the Fort Wilderness cabins are also pretty quiet, and there is a lot to do on the property. Polynesian can be romantic, but keep in mind that there is construction happening there for the new Vacation Club Villas and also on the lobby area, so that might be loud and annoying to get around.