In several regression applications, a different structural relationship might be anticipated for the higher or lower responses than the average responses. In such cases, quantile regression analysis can uncover important features that would likely be overlooked by mean regression. We develop two distinct Bayesian approaches to fully nonparametric model-based quantile regression. The first approach utilizes an additive regression framework with Gaussian process priors for the quantile regression functions and a scale uniform Dirichlet process mixture prior for the error distribution, which yields flexible unimodal error density shapes. Under the second approach, the joint distribution of the response and the covariates is modeled with a Dirichlet process mixture of multivariate normals, with posterior inference for different quantile curves emerging through the conditional distribution of the response given the covariates. The proposed nonparametric prior probability models allow the data to uncover non-linearities in the quantile regression function and non-standard distributional features in the response distribution. Inference is implemented using a combination of posterior simulation methods for Dirichlet process mixtures. We illustrate the performance of the proposed models using simulated and real data sets.