Is the following correct?
I choose $3$ irreducible in $\mathbb{Z}$.
If $g=(1+(3))T^{4} - (12 + (3))T^{2} + (64 + (3)) \in \mathbb{Z[T]}/(3)$ is irreducible, then $f=T^{4} -12T^{2} +64 \in \mathbb{Z}[T]$ is irreducible.
$g=(1+(3))T^{4} - (12 + (3))T^{2} + (64 + (3))=(1+(3))T^{4} + (1+ (3)) \in \mathbb{Z[T]}/(3)$
Now $g$ is reducible if there is an $x \in \mathbb{Z}/(3)$ such that $x^{4} + 1=0$. That implies $x^{4} = 2 \in \mathbb{Z}/(3)$. But that element doesn't exist. Hence $f \in \mathbb{Z[T]}$ is irreducible. And $f \in \mathbb{Q[T]}$ is also irreducible.