  In this paper, we first establish a narrow region principle and a decay at
infinity theorem to extend the direct method of moving planes for general
fractional $p$-Laplacian systems. By virtue of this method, we can investigate
the qualitative properties of the following Schr\"{o}dinger system with
fractional $p$-Laplacian \begin{equation*} \left\{\begin{array}{r@{\ \ }c@{\ \
}ll} \left(-\Delta\right)_{p}^{s}u+au^{p-1}& =&f(u,v), \\[0.05cm]
\left(-\Delta\right)_{p}^{t}v+bv^{p-1}& =&g(u,v), \end{array}\right.
\end{equation*} where $0<s,\,t<1$ and $2<p<\infty$. We obtain the radial
symmetry in the unit ball or the whole space $\mathbb{R}^{N}(N\geq2)$, the
monotonicity in the parabolic domain and the nonexistence on the half space for
positive solutions to the above system under some suitable conditions on $f$
and $g$, respectively.
