**Abstract** : In this paper, we construct graded tensor product characters in terms of generalized difference Macdonald raising operators which form a representation of the quantum Q-system. Characters for the graded tensor product of Kirillov-Reshetikhin modules were expressed as the constant term of a non-commutative generating function. This function is written in terms of the generators of a quantum cluster algebra, subject to recursion relations known as the quantum Q-system. The latter form a discrete non-commutative integrable system, with a set of commuting conserved quantities. In type A such conserved quantities can be treated as analogues of the Casimir elements of $U_q({\mathfrak sl}_{r+1})$. We show that the graded tensor product character is the analogue of a class I Whittaker function (to which it reduces in the Demazure case), and that the difference equations which follow from the action of the conserved quantities on characters generalize the difference quantum Toda equations of Etingof. Finally, we construct the graded characters as solutions of these equations, by introducing a representation of the quantum Q-system via difference operators which generalize the Macdonald raising difference operators of Kirillov-Noumi in the dual Whittaker limit.