We develop and evaluate SQSTS, an automated sequential procedure for computing confidence intervals (CIs) for steady-state quantiles based on the simulation analysis methods of standardized time series (STS), batching, and sectioning. Using recent theoretical developments for STS-based quantile estimation in dependent sequences, we formulate the key steps in SQSTS for controlling the growth of the batch size on successive iterations of the procedure. The variance parameter associated with the full-sample quantile estimator is estimated by a combination of estimators that are asymptotically independent of each other and the full-sample quantile estimator with increasing batch size and a fixed number of batches. Extensive experimentation revealed that SQSTS performed well compared to its competitors in terms of estimated CI coverage probabilities; and it outperformed those competitors with regard to average sample-size requirements. Finally, we outline an extension of SQSTS for computing individual CIs for a set of selected quantiles.