<p>In this paper, we propose a functional varying index coefficient quantile regression model, which can not only characterize the nonlinear interactions among a set of covariates and their influence on the response but also capture the time-dynamic features of functional/longitudinal data. Most existing semiparametric quantile regression models are special cases of the model we propose. We first develop a profile estimation method based on tensor product B-splines and establish the asymptotic properties of the profile estimators. Then, an improved estimate of the nonparametric coefficient functions is derived through spline-backfitted local linear smoothing, and its asymptotic normality is obtained. In addition, we provide a bootstrap-based testing procedure to test for the existence of interaction among covariates. The finite sample performance of the proposed method is illustrated through Monte Carlo simulations and an empirical data analysis.</p>