The Problem
While trying to preview or generate Quote Document, as a non-admin user, you observed that the pick list for the template selection is empty, like the one below:

or you have some options there, but the one you are looking for is not visible.
The Solution
The 99% cause for a missing Quote Template in the selection list is that your Quote Template status is set as In Development, and not Deployed. When the status is set to In Development, only Admin users will see the template in the dropdown menu while previewing/generating a Quote Document.
Go to your Quote Template record, and change the status.

Did it help? I hope so!