We investigate old and new q-identities in relation to universal solutions of q-difference equations, typically described by non-commuting variables in a quantum cluster algebra. We show explicitly the case of the quantum open q-Toda chain for SLN associated to the Q-system cluster algebra, and its universal series solution. The cluster algebra formulation provides a natural ``time translation" operator that commutes with the Hamiltonian of the chain. Expressing the universal solution in two different ways gives rise to q-identities that look combinatorially suggestive, although only the N=2 case is known.