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, the answer of SashaP there and a comment by GHfromMO for concrete examples: https://mathoverflow.net/questions/352837/primes-mod-4-and-integer-polynomials .