We study integrable models in the context of the recently discovered Gauge/YBE correspondence, where the Yang-Baxter equation is promoted to a duality between two supersymmetric gauge theories. We study 2d flavored elliptic genus as well as the S² partition function of the 2d N=(2, 2) quiver theories, and show that these quantities can be regarded as the partition functions of integrable lattice models living on the quiver. Our R-matrices are written in terms of theta functions, and simplifi es considerably when the gauge groups are Abelian.