Let $t$ be another variable. Then $f(X)+t$ is irreducible over $\mathbb Q(t)$. By Hilbert's Irreducibility Theorem there are infinitely many $a\in\mathbb Z$ such that $f(X)+a$ is irreducible over $\mathbb Q$. Often that is only formulated for $a\in\mathbb Q$, but the proofs which I know actually yield integers $a$.

As to irreducibility over $\mathbb Z$ versus over $\mathbb Q$, it seems to me that you should look up the Gauss Lemma.