In 16+ years working with NIST approved hardware devices I have NEVER EVER heard of NIST forcing to use a specific p and Q.
Never.
FIPS approved crypto devices exist since 20 years that, among other things like protecting keys and running application code within a security boundary, also make available a true hardware random number generator based on entropy.
I can’t see why NIST would approve such hardware devices that, to my knowledge, having to generate a true random number on demand, have no constraint whatsoever on any algorithm variable, seed or salt.
However, I’ll talk to my certification engineers to know more, promised, so you can sleep easier