With modern technology development, functional data are often observed in various scientific fields. Quantile regression has become an important statistical methodology. In this paper, we consider the estimation and inference about varying coefficients models for functional responses on quantile regression processes. We first propose to estimate the quantile smooth coefficient functions using local linear approximations, obtain the global uniform Bahadur representation of the estimator with respect to the time or the location and the quantile level, and show that the estimator converges weakly to a two-parameter continuous Gaussian process, and then we obtain asymptotic bias and mean integrated square error of smoothed individual functions and their uniform convergence rate under the given quantile level. We propose a global test for linear hypotheses of varying coefficient functions under quantile processes, and derive its asymptotic distribution under the null hypothesis; and also give their simultaneous confidence bands. For develop these inferences, some unknown error densities are estimated by the ``residual-based" empirical distributions. A Monte Carlo simulation is conducted to examine the finite-sample performance of the proposed procedures. Finally, we illustrate the estimation and inference procedures of QRVC to diffusion tensor imaging data and ADHD-200 fMRI data.