Even much stronger statements are nearly immediate consequence of Hilbert's irreducibility theorem. For example: Choose any $f_1$, $f_2$ coprime of $g$, then by Hilbert's irreducibility theorem there exist infinitely many $a\in \mathbb{Z}$ such that $$g(X)+a f_1(X) \quad and \quad g(X)+a f_2(X) $$ are both irreducible. Similarly there exist infinitely many $a\in \mathbb{Z}$ such that $$ (a f_1)^2 + 1 $$ is irreducible (this is an analogue of a problem of Landau asking whether there are infinitely many primes of the form $n^2+1$).
More generally, a Shinzel Hypothesis H version for $\mathbb{Z}[X]$ is nearly immediate from Hilbert's irreducibility theorem.
Edit: Hilbert's irreducibility theorem asserts that for every $F_1(T,X),\ldots, F_r(T,X)\in \mathbb{Q}(T)[X]$ that are irreducible, there exist infinitely many $t\in \mathbb{Z}$ such that all $F_i(t,X)$ are (defined) and irreducible in $\mathbb{Q}[X]$.