Quantiles are often used in risk evaluation of complex systems. In some situations, as in regulations regarding safety analyses of nuclear power plants, a confidence interval is required for the quantile of the simulation's output variable. In our current paper, we develop methods to construct confidence intervals for quantiles when applying Latin hypercube sampling, a variance reduction technique that extends stratification for sampling in higher dimensions. Our approaches employ the batching and sectioning methods when applying replicated Latin hypercube sampling, with a single Latin hypercube sample in each batch, and samples across batches are independent. We have established the asymptotic validity of the confidence intervals developed in this paper. Moreover, we have proven that quantile estimators from a single Latin hypercube sample and replicated Latin hypercube samples satisfy weak Bahadur representations. An advantage of sectioning over batching is that the sectioning confidence interval typically has better coverage, which we observe in numerical experiments.