Quantile regression assumes

 - the normal regression assumptions of linearity and additivity (unless you add more terms to the model)
 - independence of observations
 - very large sample size, as quantile regression is not very efficient
 - $Y$ is very continuous; quantile regression doesn't work well when there are many ties at one or more values of $Y$

You might also consider semiparametric regression (e.g., proportional odds or hazards models) which are more efficient and also allow you to estimate the mean.

My [RMS](http://fharrell.com/links) course notes goes a bit more into quantile and semiparametric regression in the chapter on ordinal models for continuous $Y$.