In this work, a methodology to monitor a shift in the quantile of a distribution that is a member of the log-symmetric family is proposed. Because the sampling distribution of a quantile estimator is often not available, the parametric bootstrap method is used to determine this sampling distribution and to establish the control limits when the process measurements follow a log-symmetric distribution. The mentioned family is helpful for describing the behavior of data following a distribution with positive support and that is skewed to the right. Monte Carlo simulations are carried out to investigate the performance of the proposed bootstrap control charts for quantiles. An application regarding failure data due to stress on carbon fibers is presented for illustration when monitoring reliability data. This illustration shows that non-conventional models, other than the Birnbaum-Saunders, log-normal and Weibull distributions, have potential to be used in practice. Two model selection procedures are considered to assess adequacy to the data. To facilitate the public use of the proposed methodology, we have created an R package named chartslogsym whose main functions are detailed in this paper.