  We show that the recently derived ($q$-) discrete form of the Painlev\'e VI
equation can be related to the discrete P$_{\rm III}$, in particular if one
uses the full freedom in the implementation of the singularity confinement
criterion. This observation is used here in order to derive the bilinear forms
and the Schlesinger transformations of both $q$-P$_{\rm III}$ and $q$-P$_{\rm
VI}$.
