Summary: ``In this article, we consider the following coupled fractional nonlinear Schrödinger system in $\Bbb R^N$ $$ \cases (-\Delta)^su+P(x)u=\mu_1|u|^{2p-2}u+\beta|v|^p |u|^{p-2}u,&x\in\Bbb R^N,\\ (-\Delta)^sv+Q(x)v=\mu_2|v|^{2p-2}v+\beta|u|^p|v|^{ p-2} v,&x\in\Bbb R^N,\\ u,\quad v\in H^s(\Bbb R^N), \endcases $$ where $N\geq2$, $00$, $\mu_2>0$ and $\beta\in\Bbb R$ is a coupling constant. We prove that it has infinitely many non-radial positive solutions under some additional conditions on $P(x)$, $Q(x)$, $p$ and $\beta$. More precisely, we will show that for the attractive case, it has infinitely many non-radial positive synchronized vector solutions, and for the repulsive case, infinitely many non-radial positive segregated vector solutions can be found, where we assume that $P(x)$ and $Q(x)$ satisfy some algebraic decay at infinity.''