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In this paper, we seek to prove the equality of the |$q$| -graded fermionic sums conjectured by Hatayama et al. [ 14 ] in its full generality, by extending the results of Di Francesco and Kedem [ 9 ] to the non-simply laced case. To this end, we will derive explicit expressions for the quantum |$Q$| -system relations, which are quantum cluster mutations that correspond to the classical |$Q$| -system relations, and write the identity of the |$q$| -graded fermionic sums as a constant term identity. As an application, we will show that these quantum |$Q$| -system relations are consistent with the short exact sequence of the Feigin–Loktev fusion product of Kirillov–Reshetikhin modules obtained by Chari and Venkatesh [ 5 ]. [ABSTRACT FROM AUTHOR]