Maximum analysis consists of modeling the maximums of a data set by con- sidering a specific distribution. Extreme value theory (EVT) shows that, for a sufficiently large block size, the maxima distribution is approximated by the gen- eralized extreme value (GEV) distribution. Under EVT, it is important to observe the high quantiles of the distribution. In this sense, quantile regression tech- niques fit the data analysis of maxima by using the GEV distribution. In this context, this work presents the quantile regression extension for the GEV distri- bution. In addition, a time-varying quantile regression model is presented, and the important properties of this approach are displayed. The parameter estima- tion of these new models is carried out under the Bayesian paradigm. The results of the temperature data and river quota application show the advantage of using this model, which allows us to estimate directly the quantiles as a function of the covariates. This shows which of them influences the occurrence of extreme temperature and the magnitude of this influence.