  Based on the works by Kajiwara, Noumi and Yamada, we propose a canonically
quantized version of the rational Weyl group representation which originally
arose as "symmetries" or the B\"acklund transformations in Painlev\'{e}
equations. We thereby propose a quantization of the discrete Painlev\'{e} VI
equation as a discrete Hamiltonian flow commuting with the action of
$W(D_4^{(1)})$.
