As the title says, is it possible?
No, this isn’t possible. There are several ways you can encode “the set of all prime numbers” as a language, and the standard ways of doing so (writing the number out in binary, writing out a number of tally marks equal to the number, etc.) aren’t regular.
Formally proving this is a bit tricky but is doable using either the pumping lemma for regular languages or the Myhill-Nerode theorem. The crux of the arguments boil down to the fact that replicating parts of prime numbers repeatedly will eventually give you a non-prime number, and that’s where the technical details of the proofs come in.
Hope this helps!