We consider a class of nonparametric quantile regression (QR) models with endogenous regressors. Building upon the semiparametric QR model in Lee (2007), we develop a nonparametric framework for quantile regression in a triangular system of equations. We provide a set of conditions under which the parameters are nonparametrically identified. Then, we propose to use the penalized sieve minimum distance (PSMD) estimation approach of Chen and Pouzo (2012) to estimate the parameters. We establish the consistency and convergence rate of the PSMD estimator. Since the identification is based on a control function approach, the PSMD estimator does not suffer from an ill-posed inverse problem. A Monte-Carlo simulation study confirms that the PSMD estimator performs well in finite samples.