  Nakajima introduced a $t$-deformation of $q$-characters, $(q,t)$-characters
for short, and their twisted multiplication through the geometry of quiver
varieties. The Nakajima $(q,t)$-characters of Kirillov-Reshetikhin modules
satisfy a $t$-deformed $T$-system. The $T$-system is a discrete dynamical
system that can be interpreted as a mutation relation in a cluster algebra in
two different ways, depending on the choice of direction of evolution. In this
paper, we show that the Nakajima $t$-deformed $T$-system of type $A_r$ forms a
quantum mutation relation in a quantization of exactly one of the cluster
algebra structures attached to the $T$-system.
