<p>The $\mathfrak{sl}_3$ colored Jones polynomial $J_{\lambda}^{\mathfrak{s}\mathfrak{l}_{3}}(L)$ is obtained by coloring the link components with two-row Young diagram $\lambda$. Although it is difficult to compute $J_{\lambda}^{\mathfrak{s}\mathfrak{l}_{3}}(L)$ in general, we can calculate it by using Kuperberg $A_2$ skein relation. In this paper, we show some formulas for twisted two strands colored by one-row Young diagram in $A_2$ web space and compute $J_{(n,0)}^{\mathfrak{s}\mathfrak{l}_{3}}(T(2,2m))$ for an oriented $(2, 2m)$-torus link. These explicit formulas derives the $\mathfrak{sl}_3$ tail of $T(2, 2m)$. They also give explicit descriptions of the $\mathfrak{sl}_3$ false theta series with one-row coloring because the $\mathfrak{sl}_2$ tail of $T(2, 2m)$ is known as the false theta series.</p>