Presumably they moved every 10 years or so, when the lack of aging was starting to get weird.
But they probably left Forks a bit earlier, I can't imagine spending 10 years in such a dirtwater place.
If you are referring to the Cullens, yes they would eventually leave. It was a plan they usually followed every few decades to move on before normal humans noticed they had not aged at all. Usually they would stay in one place for, at most, 20 years before leaving, though sometimes they would only stay for 5-10 years, depending on where they were living. I remember hearing about some of the ways they blended in with the locals.
- Sometimes Alice/Jasper or Rosalie/Emmet would go off and live as married couples in separate locations
- sometimes the coven would go on trips to visit other parts of the world, though they had to be careful about not being discovered
- in some cases, they would go live with that vampire family up in Alaska, where it's remote and they wouldn't have many neighbors
- I have no doubt the Cullens are rich enough to have several homes scattered around America that they'd cycle through
- more than once some of them would go back to college to learn new things and take night classes (Carlisle in particular would need to do so to keep his medical knowledge up to date)
- I wouldn't think it was unusual for some of them to go live in another country for a while, preferably one without a lot of sunlight throughout the year
- every few decades they would have to get new identities faked for them to use without drawing suspicion
So yeah, the Cullens, including Bella and Edward, would eventually leave Forks, but that was entirely up to them.