Traveling with 6 children ages 16+

Since your kids appear to be teens or older, I would think you would need at least 2 rooms, or even 3 - think of it: 3 bathrooms! 2 rooms would mean 2 quads. With 3 you could get a double for Mom and Dad, and 2 triples for the kids. Occupancy ratings are strictly adhered to, so you can't expect to be able to let one or two kids sleep on the floor. I like to use booking.com for research. It will give you a good idea of what is available. For what you need, you are correct to start looking now and book as soon as you can. You might even find some good deals since you are booking early. I suggest booking directly through the hotels once you find what you're looking for. December is a popular time in Europe, as you probably know.

I would look at housing rentals through Airbnb or VRBO (or whatever other search company you prefer. Homeaway?). We travel as a family of 5, and yes occupancy numbers are strictly adhered to which almost always means we'd need two rooms for the 5 of us and we find Airbnb rentals far cheaper. Try comparing.
On the other hand, maybe you are committed to staying in hotels, in which case I dont have names to recommend, obviously ;)

If your kids are over 16 you probably need multiple hotel rooms. You'll have to debate with your kids whether any of them are willing to share rooms.

An option for larger cities if you are open to hostels - it may be possible to fully book a 6-bunk room for the kids and a private for yourselves. It's much more common these days for hostels to have both privates and bunks. You would probably want to communicate directly even though you are not a "group" (usually starts at 10 for a hostel) to make sure you would get them together in one room. I could see kids enjoying that experience, and given their ages, it could be a stepping stone toward understanding what independent travel as a young adult might look like.

Where are you going in Belgium and France? And what is your budget?

Even much smaller kids are people for booking rooms and at the age of yours, everyone is clearly an adult for any pricing purposes. (except the 17 year olds will be free in museums in Paris for the most part). You need two quads or better yet 2 triples and a double and I would be looking for apartments although it is a hassle for one or two nights and hard to find big enough for 8 people. IN the US kids are free in adult rooms; this is not true in France where people have been turned away from a double with an 18 mos old. Everyone has to be booked for the room (and of course with yours no question about that -- some hotels will allow one small child in a room for no or little cost)