... my kids still nap. My son (who is 4), naps everyday otherwise he turns into a TROLL.... by late afternoon.
My daughter who is 8, will nap if/when tired.
I have always taught them that napping... is not a 'bad' thing.... it is what their body needs, when tired. And adults nap too.
So I don't make it like a punishment/negative thing for them.
But yes, every kid is different per naps.
For me as well, in the afternoon is it quiet time, because my son naps. Then my daughter who is older, will have quiet time and do her homework etc. AND... I also say that this time is also "Mommy's time"... for nap or doing work at home or whatever. I also, use that nap time for my son, to do 'bonding' with my daughter. But it is quiet things, since my son is napping.
But whatever it is.... we always have a routine.... everyday. They know it like autopilot, so there is rarely any 'battling' about it. All I have to do now, is verbally cue them, about it. Ahead of time.
For you, quiet time seems to be, with your daughter IN her room, by herself. For any 3 year old, they cannot, developmentally to me.... just be by themselves tinkering around for who knows how long. For them 1/2 hour can be like an eternity. And then they also want company.... with you. So, it is never a long concentrated time of them just tinkering away by themselves in their room. Alone. Nor quietly. Not until they are much older, can they do that.
Tell her maybe, that MOMMY is quiet time too. Then do it together. You could also just lay on the sofa, getting some time off your feet/closing your eyes.... while in her room... and she may doze too. I have done that, and it worked.
But if she is really done with naps... then well, that is that.
But have a regular routine about quiet time... but not as restrictive as her being IN her room for the whole time. That is arduous for any 3 year old.
I had my son when my daughter was 3 going on 4 years old. Ever since he was an infant, and per his naps.... I coordinated that with my daughter's naps too, synchronizing it. Same timing. Then as my daughter got older... she just got used to her brother napping, then it was soft-voice time and quite time... and it was just routine. Both my kids KNOW... that when someone is napping.... they need to be quiet and use soft-voices. It is quiet time. And do quiet things. They do so. They understand.
all the best,
Susan