I use MCMC to study a distribution. Thanks to the modified Central Limit Theorem found in Robert and Casella's Monte Carlo Statistical methods, one can approach the expectation of the distribution, its variance, actually any integral of the shape $\int g(x) f(x) dx $, $f(\cdot)$ being the density function and $g(\cdot)$ being the function I want to integrate.
However, I do not think that the quantiles of the distribution are an integral like this one. So does it take another result to estimate quantiles from MCMC ? Is there an integral form of quantile estimator that I did not spot ? Or is it impossible ?