We propose SQSTS, an automated sequential procedure for computing confidence intervals (CIs) for steady-state quantiles based on Standardized Time Series (STS) processes computed from sample quantiles. We estimate the variance parameter associated with a given quantile estimator using the order statistics of the full sample and a combination of variance-parameter estimators based on the theoretical framework developed by Alexopoulos et al. in 2022. SQSTS is structurally less complicated than its main competitors, the Sequest and Sequem methods developed by Alexopoulos et al. in 2019 and 2017. Preliminary experimentation with the customer delay process prior to service in a congested M/M/1 queueing system revealed that SQSTS performed favorably compared with Sequest and Sequem in terms of the estimated CI coverage probability, and it significantly outperformed the latter methods with regard to average sample-size requirements.