Currently, working with partially observed functional data has attracted a greatly increasing attention, since there are many applications in which each functional curve may be observed only on a subset of a common domain and unobserved elsewhere, and the incompleteness makes most existing methods for functional data analysis ine_ective. In this paper, motivated by the appealing characteristics of conditional quantile regression, we consider the functional linear quantile regression, assuming the explanatory functions are observed partially on dense but discrete point grids of some random subintervals of the domain. A function principle component analysis (FPCA) based estimator is proposed for the slope function, and the convergence rate of estimator is investigated. In addition, the _nite sample performance of the proposed estimator is evaluated through simulation studies and a real data application.