This paper proposes a moments-based approach to the identication and estimation of panel data quantile regression (QR) models with fixed effects when the number of time periods T is small. When the covariates have discrete support and fixed effects are pure location shifts,I show that the QR model is identied and suggest an estimator based on the recovering the distribution function from a sequence of its moments. When the covariates are continuously distributed, I show that the QR model can be identied even when xed eects are allowed to vary across quantiles.