  We define and construct a quantum Grothendieck ring for a certain monoidal
subcategory of the category $\mathcal{O}$ of representations of the quantum
loop algebra introduced by Hernandez-Jimbo. We use the cluster algebra
structure of the Grothendieck ring of this category to define the quantum
Grothendieck ring as a quantum cluster algebra. When the underlying simple Lie
algebra is of type $A$, we prove that this quantum Grothendieck ring contains
the quantum Grothendieck ring of the category of finite-dimensional
representations of the associated quantum affine algebra. In type $A_1$, we
identify remarkable relations in this quantum Grothendieck ring.
