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　the　incompleteness　makes　most　existing　methods　for　functional　data　analysis　ineffective.　In　this　paper,　motivated　by　the　appealing　characteristics　of　conditional　quantile　regression,　the　authors　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　functional　principal　component　analysis　(FPCA)　based　estimator　is　proposed　for　the　slope　function,　and　the　convergence　rate　of　the　estimator　is　investigated.　In　addition,　the　finite　sample　performance　of　the　proposed　estimator　is　evaluated　through　simulation　studies　and　a　real　data　application.