Sets of Prime Numbers Generated By an Irreducible Monic Polynomial

The answer is no. See Theorem 3 in Prime Numbers in Certain Arithmetic Progressions:

If $f,g \in \mathbb{Z}[x]$ are non-constant, then $P(f) \cap P(g)$ is infinite.

The proof there is quite short so I might as well include it:

Suppose $\alpha$ is a root of $f$ and $\beta$ is a root of $g$. By Dedekind 's theorem, except for a finite number of exceptions, $p$ is a prime divisor of $f$ exactly when $p$ has a first degree prime ideal factor in the field $\mathbb{Q}(\alpha)$. A similar statement holds for $g$. Consider the field, $\mathbb{Q}(\alpha, \beta)=\mathbb{Q}(\theta)$ for some $\theta \in \mathcal{O}_K$. If $h$ is the minimal polynomial of $\theta$ then Schur's Theorem assures us that there are infinitely many primes which have a first degree prime ideal factor in $\mathbb{Q}(\alpha, \beta)$. These primes thus have a first degree prime ideal factor in both $\mathbb{Q}(\alpha)$ and $\mathbb{Q}(\beta)$. Thus, except for a finite number of exceptions, these primes are in both $P(f)$ and $P(g)$.