  We study motivic zeta functions for $\mathds{Q}$-divisors in a
$\mathds{Q}$-Gorenstein variety. By using a toric partial resolution of
singularities we reduce this study to the local case of two normal crossing
divisors where the ambient space is an abelian quotient singularity. For the
latter we provide a closed formula which is worked out directly on the quotient
singular variety. As a first application we provide a family of surface
singularities where the use of weighted blow-ups reduces the set of candidate
poles drastically. We also present an example of a quotient singularity under
the action of a nonabelian group, from which we compute some invariants of
motivic nature after constructing a $\mathds{Q}$-resolution.
