Is it desirable / possible to apply ensemble learning methods (boosting, bagging, etc) to the quantile regression problem?
Yes, and in fact this has been addressed in the literature. Quantile random forests adapt the random forest framework for quantile regression and the author provides consistency proofs. There is an R package implementing the technique as well here.
In terms of desirability, ensembles will help with the bias/variance problem for quantile regression just as well as predicting the mean.