We study a new integrable probabilistic system, defined in terms of a stochastic colored vertex model on a square lattice. The main distinctive feature of our model is a new family of parameters attached to diagonals rather than to rows or columns, like in other similar models. Because of these new parameters the previously known results about vertex models cannot be directly applied, but nevertheless the integrability remains, and we prove explicit integral expressions for $q$-deformed moments of the (colored) height functions of the model. Following known techniques our model can be interpreted as a $q$-discretization of the Beta polymer model from arXiv:1503.04117 with a new family of parameters, also attached to diagonals. To demonstrate how integrability with respect to the new diagonal parameters works, we extend the known results about Tracy-Widom large-scale fluctuations of the Beta polymer model.