You can apply the standard within transformation to your data
manually by deducting the mean (or median) values of the dependent
and independent variables for each panel unit from all
observations. This will eliminate the panel fixed effects. You can
recover the fixed effects in the usual way from the average or median
values of the residuals for each panel unit.
However, you should think quite carefully about what it is that you
are estimating. There is a substantial literature among those
interested in quantile regression about what the precise meaning of
the conditional quantiles generated by these models - for example
read the textbooks by Cameron & Trivedi or Wooldridge. By removing
the fixed effects you would be estimating conditional median
deviations from the mean/median values of the observed
variables. This may or may not be what you are trying to obtain.
