\(x+y+z=0\Rightarrow\left\{{}\begin{matrix}x+y=-z\\y+z=-x\\z+x=-y\end{matrix}\right.\)
\(Q=\left(\frac{x+y}{y}\right)\left(\frac{y+z}{z}\right)\left(\frac{z+x}{x}\right)=\frac{\left(-z\right).\left(-y\right)\left(-x\right)}{xyz}=\frac{-xyz}{xyz}=-1\)