  We introduce a new set of $q$-difference operators acting as raising
operators on a family of symmetric polynomials which are characters of graded
tensor products of current algebra ${\mathfrak g}[u]$ KR-modules \cite{FL} for
${\mathfrak g}=A_r$. These operators are generalizations of the Kirillov-Noumi
\cite{kinoum} Macdonald raising operators, in the dual $q$-Whittaker limit
$t\to\infty$. They form a representation of the quantum $Q$-system of type $A$
\cite{qKR}. This system is a subalgebra of a quantum cluster algebra, and is
also a discrete integrable system whose conserved quantities, analogous to the
Casimirs of $U_q({\mathfrak sl}_{r+1})$, act as difference operators on the
above family of symmetric polynomials. The characters in the special case of
products of fundamental modules are class I $q$-Whittaker functions, or
characters of level-1 Demazure modules or Weyl modules. The action of the
conserved quantities on these characters gives the difference quantum Toda
equations \cite{Etingof}. We obtain a generalization of the latter for
arbitrary tensor products of KR-modules.
