  This paper is concerned with a quasilinear Schr\"{o}dinger system in $\mathbb
R^{N}$ $$\left\{\aligned &-\Delta
u+A(x)u-\frac{1}{2}\triangle(u^{2})u=\frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},\\
&-\Delta
v+B(x)v-\frac{1}{2}\triangle(v^{2})v=\frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,\\
& u(x)\to 0\ \hbox{and}\quad v(x)\to 0\ \hbox{as}\ |x|\to
\infty,\endaligned\right. $$ where $\alpha,\beta>1$ and
$2<\alpha+\beta<\frac{4N}{N-2}$ ($N \geq 3$). $A(x)$ and $B(x)$ are two
periodic functions. By minimization under a convenient constraint and
concentration-compactness lemma, we prove the existence of ground states
solution. Our result covers the case of $\alpha+\beta\in(2,4)$ which seems to
be the first result for coupled quasilinear Schr\"{o}dinger system in the
periodic situation.
