Yes, well fed cats do hunt because it is an inherited, instinctive trait but a domestic cat may not kill prey quite so often nor eat prey as consistently if they are well fed on high quality food. Being well fed may contribute to a decreased need to hunt.
Well fed – still hunt
Keeping pet cats well fed probably does not make a great of difference in altering feline behavior to save wildlife from being hunted. Although in some studies, carried out in 1979, 1998 and 2011 it was found that hunting by domesticated, owned cats can be linked to their body condition or to the amount of meat in their diet and hunger (Biben 1979; Robertson 1998; Silva-Rodríguez and Sieving 2011). In one study in 2019, inadequate feeding of pet cats whose role was to control vermin on Polish farmsteads was linked to high predation rates on mammals and birds. But this is the other side of the coin: underfeeding equates to more predation. Does overfeeding equate to less predation?
However, any benefit that may be accrued by ensuring that your cat is well fed on a high-quality diet is probably undermined by the opportunities available to hunt which is a major determining factor.
Cats living closer to natural habitats compared to those in heavily urbanised areas have higher predation rates. In short, they kill more animals. For example, in two studies in Canberra and Adelaide, Australia, the predation rates of domestic cats in households increased in line with their proximity to woodlands (Paton 1991; Barratt 1997b). To put it another way, when pet cats, living in nice homes, also live near natural hunting habitats where there’s lots of prey they will hunt there, if allowed to do so, no matter whether they’re well fed or not.
So what conclusion can be drawn from this? Domestic cats are programmed by their inheritance to respond to the sounds and movements that prey animals make and they will react instinctively to them, track them down, stalk, attack and kill the animal. Whether they then eat prey may partly depend upon how well fed they are. If they are well fed perhaps to the point where they become obese (and there is a feline obesity problem in developed countries) it may slow them down both in terms of desire and their physical abilities. Therefore perhaps it might be possible to say that well fed cats hunt less determinedly and kill slightly less wild animals as a consequence. But the fundamental hard-wired hunting desires are unaffected.
Some pages on feline obesity