We study a quantum version of the $n$-dimer model from statistical mechanics, based on the formalism from quantum topology developed by Reshetikhin and Turaev (the latter which, in particular, can be used to construct the Jones polynomial of a knot in $\mathbb{R}^3$). We apply this machinery to construct an isotopy invariant polynomial for knotted bipartite ribbon graphs in $\mathbb{R}^3$, giving, in the planar setting, a quantum $n$-dimer partition function. As one application, we compute the expected number of loops in the (classical) double dimer model for planar bipartite graphs.