Can you cook quince whole?
No, this should work. 

But you need to get that precise point between too hard and too soft to cut. And during cooking the fruit will soften from the outside inwards: You might find yourself handling the fruit during cooking, so either you have to touch hot fruit or interrupt the cooking, let cool, peel, cook again....

If you are aiming for quince puree, just rub off the fuzz, cook the fruit and then push it through a food mill or sieve. You'll need a pretty large pot though, if you want to cook whole fruits vs. smaller pieces.

On the other hand, if you have a reasonable sharp and heavy chef's knife and a sharp paring knife, I suggest you try cutting a raw quince first. You might find it less difficult than it's often said.