  We propose solutions of the quantum Q-systems of types $B_N,C_N,D_N$ in terms
of $q$-difference operators, generalizing our previous construction for the
Q-system of type $A$. The difference operators are interpreted as $q$-Whittaker
limits of discrete time evolutions of Macdonald-van Diejen type operators. We
conjecture that these new operators act as raising and lowering operators for
$q$-Whittaker functions, which are special cases of graded characters of fusion
products of KR-modules.
