Before revision:  The  first-stage is analogue to the least squares case, i.e., a conditional mean regression of the endogenous variables on the instruments, with the difference that for the QR case is a weighted regression.
After revision:  The quantile first-stage is analogue to the least squares case, i.e., a conditional mean regression of the endogenous variables on the instruments, with the difference that for the QR case is a weighted regression.