No. The set of subsets of $P$ satisfying the OP conditions is of  cardinality continuum while the set of polynomials  with integer coefficients is countable. This answer gives implicite examples. But see [this question][1], the answer of SashaP there and a comment by GHfromMO for concrete examples.


  [1]: https://mathoverflow.net/questions/352837/primes-mod-4-and-integer-polynomials