So far all of my kids have been good for naps except my almost 4 year old. That's understandable as I see my own almost 3 year old starting to fight off naps more and more each day.
On days I have non-nappers I have them go down to the play room and play with 'quiet toys.' I will put on the tv and they catch watch, play with dolls or read books ...that's it. I don't really require them to lay quietly as I've tried that and it was more of a hassle than anything. I enjoy my quiet time as well and with trying to get the kid(s) to lay down and stay required me to miss out on my own time.
So to answer your question, I would say by the age of 3 I no longer require them to nap or lay down for quiet time.