This is not good English. We must say either "He is not moving" or "He is not cahnging apartments" or "He is not moving from his apartment" or something like that. "He is not moving apartments" would mean that he is not putting a number of apartments on a moving van and transporting the rooms to another location.
This sentence is about moving between apartments, from one and into the other. The concept is quite similar to הוא מחליף חולצה, "he is changing his shirt". If you want to say he is moving out, you can say הוא עוזב את הדירה or maybe הוא עובר מהדירה, but that is less frequent. But you can say הוא עובר לדירה, he is moving in.
The normal way to express this in English is "he is not moving", and context tells you that this is about going from living in one place to living in another, rather than about standing still like a statue.
If you said your sentence to an English speaker, they would have no idea what you're saying.