Quantile regression has received a great deal of attention as an important tool for modeling statistical quantities of interest other than the conditional mean. Varying coefficient models are widely used to explore dynamic patterns among popular models available to avoid the curse of dimensionality. We propose a support vector quantile regression model with varying coefficients and its two estimation methods. One uses the quadratic programming, and the other uses the iteratively reweighted least squares procedure. The proposed method can be applied easily and effectively to estimating the nonlinear regression quantiles depending on the high-dimensional vector of smoothing variables. We also present the model selection method that employs generalized cross validation and generalized approximate cross validation techniques for choosing the hyperparameters, which affect the performance of the proposed model. Numerical studies are conducted to illustrate the performance of the proposed model.