Linear models are models like $\mathbb E\left[Y\vert X\right]=X\beta$.
Linear quantile regression models replace $\mathbb E\left[Y\vert X\right]$ with $Q_{\tau}\left(Y\vert X\right)$ for conditional quantile $\tau$, giving $Q_{\tau}\left(Y\vert X\right)=X\beta$.
Generalized linear models include a link function to transform the conditional expected value of a linear model, so $g\left(\mathbb E\left[Y\vert X\right]\right)=X\beta$.
Is there any role for generalized quantile regression that transforms the conditional quantile $\tau$, like $g\left(Q_{\tau}\left(Y\vert X\right)\right)=X\beta$? How would we go about developing a likelihood function to maximize or a loss function to minimize?