How to cut chicken without producing loud noises

Plan a bit ahead. Move the chicken from the freezer to the fridge the night before, so it's thawed (or at least not rock-hard) when it's time to cut it. Or if for some reason that won't work, thaw it at least partially under running water.

Beyond that, make sure your knife is sharp, so you're not having to spend more effort than you should. With thawed meat and a sharp knife, you shouldn't need violent swings of your knife.

To answer the actual question you asked, a kitchen towel (or several layered) will dampen impact on the counter some. And you might find that different parts of the counter rattle less, or even try cutting on your table. But I'd really suggest avoiding having to whack it hard enough to make the counter shake in the first place!

...cut up the chicken before you freeze it, and portion it out into meal-sized portions at that time, rather than freezing a whole chicken and hacking it up while frozen?

Or, (perhaps you are buying whole frozen chickens?) a meat saw of some sort, but that seems a bit absurd for chicken - still, a lot quieter if cutting it while frozen solid. Perhaps a well-scrubbed coping saw would be a nice chicken-sized solution for this.

You might think about the chisel solution. When I have to cut something really hard, I sometimes set my knife where I want it to go, and use my pestle to tap down on the back of the knife, and slowly tap the knife through what I'm cutting. Less force wasted through swings and noise, more concentrated on the actual area I'm cutting through. You will still get some noise, which might be muffled by towels, but it might help. Of course, if you're lacking a nice granite mortar and pestle set like mine, you might use a hammer for the same effect, or a clean stone, or anything sturdy and heavy and controllable (controllable is key, you want measured taps, not sqwushed fingers). You can even pick up an actual chisel (and hammer) for food purposes, if this is a regular problem.

Another thing that might work is placing your knife, and leaning into it (setting your weight behind your blade). You can rock the blade back and forth, to work it deeper, let it stand a while and return to it (not cutting through in a single blow). Your weight concentrated on the edge of the blade can do quite a bit of your cutting for you - and since it is consistent pressure, it shouldn't make much noise at all. Obviously, you want to be careful - if the blade slides sideways, or twists in your hands, or if the food splits when you're not expecting it, it can be dangerous. You should lean in gently at first, until you're sure the blade is firmly bitten into the food, make sure everything is clear of the blade except what you want to be cut. And take breaks to rock the blade, perhaps slice at the groove a bit, and check your progress (less pressure as you get closer to cut through) - so that you're not taken by surprise when the food splits apart and the knife has no opportunity to try to twist free.

Alternately, you might find that slicing rather than chopping will help - slicing makes little noise since the force is sideways, and consistent rather than sudden. It will take a bit more time - you're depending on consistent force (the pressure of your hand) instead of a sudden shock like chopping, and it will take several passes for the knife to properly slice through something . Slicing relies more on the sharpness of the knife than vertical pressure, a slicing motion can be quite gentle if the blade is sharp, but you have to balance the downward pressure of your hand vs the sharpness of your blade vs the toughness of what you're cutting.

Both slicing and leaning into the cut work better when the food is consistent in texture - something like a bone in your chicken may hang it up and not work as well when you hit it, you might have to just chop through that. But, they might help you get through the rest of the cutting, so that you only need the one bit chopped through by force (or tapped through, chisel fashion).