Nap time at my sons Montessori seems to be voluntary.
In his 3-6 room there are 25 kids. There are 4 nappers, my son included (he is 3). After lunch the nappers go to this alcove in the room and nap on their mats. His teachers just instilled this, this year; having nappers nap in the same room as the non-nappers. They said part of the M. thing was to have all children working together so they wanted the nappers to stay in the classroom. The other children learn respect and consideration (using quiet voices) while the nappers nap.
My son used to bring a little car to school and went to sleep holding that. In fact, it seems that all the children can bring one thing to school so long as they keep it in their cubby and it does not distract them from work. At the beginning of the year the teachers and I discussed my sons sleeping needs - he works better with a nap. The teachers worked with the nappers (set out their "beds", patted them on their backs to help them sleep, dimmed the lights in the alcove). Now going to nap after lunch is routine for the 4 nappers. The teachers always have to wake him up as he is still sleeping by closing time.
At anyrate, I think no kids in his school are forced to lay down for quiet time. The other 20 kids continue with work and activities in the afternoon as they do in the morning.
Previous to this year, though, the school used to have a nap room and an adult to supervise. After lunch the nappers from the 4 different rooms were taken to this other carpeted room. And they napped there, which allowed the non-nappers to continue their activities.
It seems your school is a bit different as they have quiet time for everyone.
I would be frustrated too, I think "forcing" nap time is going against the bigger picture of the M. philosophy. Either way, it is too bad they are not flexible to allow your child to read a book instead.
I don't think the no book @ quiet time is a policy. Nor do I think quiet time, on a mat, is a global policy.
(My sons M. is affiliated with AMS, AMI, and a few other abbreviations)