No – this is an urban myth. One type of cell (or substance for that matter) cannot turn into another. A fat cell is a fat cell, and a muscle cell is a muscle cell. Muscle does not turn into fat.
The reason this myth is so prevalent is because the external symptoms seen when a bodybuilder stops working out seem to agree with the idea that muscle turns into fat. If you work out for long enough you will build muscle mass through weight lifting and lose fat through cardio and a healthy diet. If you then stop this routine, a number of things will happen. The muscles you built will no longer be of any use to your body and it will begin to break it down and absorb it. The body does not keep excess muscle as it is just extra weight and requires extra energy. You will also likely start to gain fat as you are not doing regular cardio. If you start eating badly in addition you will gain fat at a faster pace. This may even happen rather naturally, as bodybuilders tend to become depressed when they stop a regular workout routine. This depression may result in an increased appetite. This in turn will cause an increased food intake and thereby a faster fat gain. This positive feedback loop is difficult to get out of and the best way to avoid it is to not stop working out.
We can see that the above processes in combination may look like the muscle is ‘turning into’ fat when you stop a workout routine. However, this is not the case, as there are two simultaneous and independent processes going on – the loss of muscle and the increase in body fat percentage.