Explicit cluster multiplication formulas for the quantum cluster algebra of type $A_2^{(1)}$

Let $Q$ be an affine quiver of type $A_2^{(1)}$. We explicitly construct the cluster multiplication formulas for the quantum cluster algebra of $Q$ with principal coefficients. As applications, we obtain: (1)\ an exact expression for every quantum cluster variable as a polynomial in terms of the quantum cluster variables in clusters which are one-step mutations from the initial cluster; (2)\ an explicit bar-invariant positive $\mathbb{ZP}$-basis.