To me, it is just a reflection of their age... and a transition. Not that they don't need a nap all together. Because, on some days they WILL still need a nap and be tired/fussy. SO you have to gauge it... and go by the cues of your child.
I have 2 kids. At that age, sleep naturally becomes harder for them. Not that they are not tired, but they are changing in development. So for my kids, I still kept to a nap routine, everyday. Or at the least, a quiet time. But as of now and my kids current ages... they STILL nap. My daughter on occasion (and she doesn't battle about it), and my son is napping NOW... and he is 3.5 years old. He still naps everyday. I don't force it... but it is the routine, and he does get genuinely tired. I know his cues. He even gets into the nap routine himself. My daughter too, when she needs one.
OR, you tell your kid... it is MOMMY'S nap time... and that you both lie down. And then do so. Make a routine of it. It can be in her room or on a sofa. Someplace quiet and darkened. That is what I do sometimes too... I reverse it.. and I tell them MOMMY is going to nap... and we ALL do so. And it works out for us.
My kids at that age for some reason, didn't want to nap in their bed either... so they would fall asleep on the floor of the living room or on the floor of their room. So be it. Main thing is they NAP. And they were comfortable. Then I turn off all the phones.
It was a phase, then they went back to napping in their beds.
They do this at that age.
But don't get in the habit of driving around in the car just to make her nap. That is a real ball and chain of a habit to keep up. My friend has to do that... and its ridiculous. They spend so much time, driving around, just for their tired & fussy kids to nap.
And no, you can't just expect to close the door and she amuse herself at that age and then fall asleep. You need to be there too.. nearby, in a chair, and tell her no talking and Mommy is going to "read" and then do so. Making it all quiet. But tell her to lay down....
These are some of the things I do, and my kids NEVER battle about naps. It is our routine, since day one. And I don't force it... it is just a part of the day.
Or, just make her nap time a bit later... since she is now 2 years old. In the afternoon.
Also, my son won't/can't nap, if he has to poop. Once he poops, he naps fine. After lunch and a full tummy, I know within 1 hour he will poop. Then its nap time.
About "quiet time".... my kids know that it is also "Mommy's" quiet time too... and I do quiet things like being here on the computer and doing my desk work. You make it a routine and a habit and a everyday occurrence.
We are all in the same room together, but we all do quiet things "together." It works for us.
all the best,
Susan