Quantile regression has a natural extension to generative modelling by leveraging a stronger pointwise convergence than in distribution. While the pinball quantile loss works well in the scalar case, it cannot be readily extended to the vector case. In this work, we propose a multivariate quantile approach for generative modelling using optimal transport with provable guarantees. Specifically, we suggest that by optimizing smooth functions parameterized by neural networks with respect to the dual of the correlation maximization problem, the function uniformly converges to the optimal convex potential. Thus, we construct a Brenier map as our generative quantile network. Furthermore, we introduce conditioning by approximating the convex potential using a first-order approximation with respect to the covariates. Through extensive experiments on synthetic and real datasets for conditional generative and probabilistic time-series forecasting tasks, we demonstrate the efficacy and versatility of our theoretically motivated model as a distribution estimator and probabilistic forecaster.