  For an unpunctured marked surface $\Sigma$, we consider a skein algebra
$\mathscr{S}_{\mathfrak{sl}_{3},\Sigma}^{q}$ consisting of
$\mathfrak{sl}_3$-webs on $\Sigma$ with the boundary skein relations at marked
points. We construct a quantum cluster algebra
$\mathscr{A}^q_{\mathfrak{sl}_3,\Sigma}$ inside the skew-field
$\mathrm{Frac}\mathscr{S}_{\mathfrak{sl}_{3},\Sigma}^{q}$ of fractions, which
quantizes the cluster $K_2$-structure on the moduli space
$\mathcal{A}_{SL_3,\Sigma}$ of decorated $SL_3$-local systems on $\Sigma$. We
show that the cluster algebra $\mathscr{A}^q_{\mathfrak{sl}_3,\Sigma}$ contains
the boundary-localized skein algebra
$\mathscr{S}_{\mathfrak{sl}_{3},\Sigma}^{q}[\partial^{-1}]$ as a subalgebra,
and their natural structures, such as gradings and certain group actions, agree
with each other. We also give an algorithm to compute the Laurent expressions
of a given $\mathfrak{sl}_3$-web in certain clusters and discuss the positivity
of coefficients. In particular, we show that the bracelets and the bangles
along an oriented simple loop in $\Sigma$ have Laurent expressions with
positive coefficients, hence give rise to quantum GS-universally positive
Laurent polynomials.
