When you load in QGIS a shapefile without a PRJ file, you can set the coordinate system, but no PRJ file is created.
This older question: How to obtain .prj files in QGIS?
has an answer saying menu
"Vector - Data Management Tools - Define Current Projection" does it, but it is not present in recent versions. Any suggestion of how to do it?
Of course, I can do it manually, copying the PRJ file of some other shape, but probably there is a way of doing it from within QGIS menus.