We propose an online inference method for censored quantile regression with streaming data sets. A key strategy is to approximate the martingale-based unsmooth objective function with a quadratic loss function involving a well-justified second-order expansion. This enables us to derive a new online convex function based on the current data batch and summary statistics of historical data, thereby achieving online updating and occupying low storage space. To estimate the regression parameters, we design a novel majorize-minimize algorithm by reasonably constructing a quadratic surrogate objective function, which renders a closed-form parameter update and thus reduces the computational burden notably. Theoretically, compared to the oracle estimators derived from analyzing the entire raw data once, we posit a weaker assumption on the quantile grid size and show that the proposed online estimators can maintain the same convergence rate and statistical efficiency. Simulation studies and an application demonstrate the satisfactory empirical performance and practical utilities of the proposed online method.