Tuples of polynomials over finite fields with pairwise coprimality conditions

Let $q$ be a prime power. We estimate the number of tuples of degree bounded monic polynomials $(Q_1,\dots ,Q_v)\in ({\mathbb{F}}_q[z]{)}^v$ that satisfy given pairwise coprimality conditions. We show how this generalises from monic polynomials in finite fields to Dedekind domains with finite norms.