The comments and the answer are related to the probably hopeless Bunyakovsky conjecture. It seems to me that Stefan Kohl had a different idea in mind, maybe something like the following: Let $A_p$ be the set of polynomials $f\in\mathbb Z[X]$ of degree $\ge2$ with $p\in f(\mathbb Z)$. The question amounts to asking if there is an infinite set $P$ of primes such that $\cap_{p\in P}A_p$ is not empty.

So in this form the question is if some density argument (with respect to some measure on $\mathbb Z[X]$) could be strong enough.