This paper describes Bayesian estimation of the parameters of the generalized extreme value (GEV) model with covariates. For this model the parameters of the GEV distribution are functions of covariates, allowing for dependent parameters and/or trends. A Markov chain Monte Carlo (MCMC) algorithm is generally used to estimate the posterior distributions of the parameters in a Bayesian framework. In this paper, the birth-death MCMC (BDMCMC) procedure is developed in order to carry out both parameter estimation and Bayesian model selection. The BDMCMC methods allow the jump between models of different dimensions. The general algorithm consists of two types of sampling steps. The first one involves dimension-changing moves, and the second is conditional on a fixed model. Parameters are estimated in a fully Bayesian framework, and the model is selected by the length of time that the MCMC chain remains in that model. Real and simulated data sets illustrate the usefulness of the proposed methodology.