So as a veterinarian that deals with turtles and tortoises on a regular basis, I can tell you that one of the most common things I hear from clients bringing in their pets is that the turtle has stopped eating. There are many reasons why this can happen. A turtle can stop eating as secondary for many different illnesses, but there are a few that we see more commonly.
The first thing would be that the husbandry of the turtle is incorrect. The word, husbandry, just means how you care for your turtle. What temperatures do you keep it at? What food you're offering it? What temperatures it's kept at? Does it have a source of ultraviolet light?
I think one of the biggest things that confuses people is that turtles do need a source of heat to regulate their own body temperature. So a turtle that is kept at a temperature that is too low, will actually have a slowing of its intestinal transit time. Basically, this means the turtle is going to be lethargic, and it's not going to want to eat.
Another thing I see commonly is that people aren't offering the appropriate diet. So again, it's very important that you research your individual species of turtle and figure out what it should be fed in captivity. There are many types of commercial foods, but some turtles don't take to certain types of pellets and require a more specialized diet.
So another thing that I see commonly in practice is that the turtle has not been provided with a source of ultraviolet light. What this causes is metabolic bone disease. It's important that they have the ultraviolet light to synthesize Vitamin D so that they can utilize calcium. So when turtles are calcium deficient, it causes them to lose their appetite, slow down. It also suppresses their immune system and makes them more susceptible to infections with bacteria, fungal organisms, and parasites.
So there are actually many, many different reasons why a turtle will stop eating, but it is not a normal behavior unless your turtle is hibernating. When you're keeping a turtle indoors, you really shouldn't be hibernating your turtle unless you're a breeder. So we recommend maintaining the temperatures and the lighting year round. So if your turtle stops eating for some reason, that's something to be concerned about, and I would recommend contacting a reptile veterinarian.