We construct bar-invariant $\mathbb{Z}\left[q^{\pm 1/2}\right]$-bases of the quantum cluster algebra of the valued quiver $A_2^{\left(2\right)}$, one of which coincides with the quantum analogue of the basis of the corresponding cluster algebra discussed in P. Sherman, A. Zelevinsky: Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Moscow Math. J., 4, 2004, 947–974.