Lots of good suggestions here, but I'll add my thoughts just for grins. As others have noted, the quietest restaurants generally are the Signatures at resorts, and they usually do get even quieter the later the reservation. That being said, the only truly quiet dining we've found at Disney is Victoria & Albert's. It's very pricey but an exceptional experience. Beyond that, here are my recommendations.
Artist Point: Perhaps the loveliest ambiance of the Signatures. We prefer to make an early reservation and ask for a window seat, and we've had excellent success. Generally speaking, AP is one of the least crowded Signatures, as well, perhaps due to it being off the beaten path and its reputation for having a fare such as buffalo and salmon (though it has much more than those two items). If you do go a bit later, I'd still ask for a window seat and be willing to wait a bit to secure one. Being able to look out over the pool and courtyard area adds an extra layer of uniqueness. The food is good, and the service usually exemplary. Moreover, the overall serenity of the Lodge extends to AP and makes it an exceptional place to connect with your spouse or significant other.
Jiko: A close second to AP in ambiance and perhaps even better menu. The staff is extremely courteous and helpful, and the styling inside the restaurant reflects tranquility. As with Artist Point, you can have lovely, quiet conversations with your spouse and feel as though you're being treated as a special guest.
Il Mulino: Upscale and a bit busier than either AP or Jiko, this Italian eatery has a varied menu that allows you to order small plates from their selections of pasta which is a great way to save money. The service is very good as is the food. You can still have quiet moments alone, but the turn over is greater and tables a tad closer. Still, I think yo'd have a great meal here.
Sanaa: Very laid back vibe here, and the food is quite good overall (I especially recommend the bread service--fit for two!). Very convivial and efficient service, and as a bonus, if you go during daylight hours, you can see animals on the savannah. There will be more families about, but we've never had a loud experience there.
CA Grill: Considered one of the better Signatures, it has the glamor of being atop CR where you get views of MK and Wishes at night. However, one major caveat here is that it always seems to be busier and noisier than other Signatures, perhaps due to it also being a bar and the popularity of the location. While I've had good to great meals there, I wouldn't choose it for intimacy. Joyous and celebratory dining? Yes. Quiet or romantic? Not so much.