In this paper, we focus on the partially linear varying-coefficient quantile regression model when the data are right censored and the censoring indicator is missing at random. Based on the calibration and imputation methods, a three-stage approach is proposed to construct the estimators of the linear part and the nonparametric varying-coefficient function for this model. At the same time, we discuss the variable selection of the covariates in the linear part by adopting adaptive LASSO penalty. Under appropriate assumptions, the asymptotic normality of the proposed estimators is established, and the penalized estimators are proven to have the oracle property. Simulation study and a real data analysis are conducted to evaluate the performance of the proposed estimators.