$Q=\frac{x^4}{xy+zz}+\frac{y^4}{zx+xy}+\frac{z^4}{xy+yz}+\frac{y^2z^2}{yz+z^2}+\frac{z^2x^2}{zx+x^2}+\frac{x^2y^2}{xy+y^2}$
$\ge \frac{(x^2+y^2+z^2)^2}{2(xy+yz+zx)}+\frac{(xy+yz+zx)^2}{x^2+y^2+z^2+xy+yz+zx}$
$=\frac{t^2}{2}+\frac{1}{t+1} \ge \frac{t}{2}+\frac{1}{t+1}=\frac{t+1}{4}+\frac{1}{t+1}+\frac{t-1}{4} \ge 1$
Với $t=x^2+y^2+z^2 \ge xy+yz+zx=1$
$\Rightarrow \min Q=1\Leftrightarrow x=y=z=\sqrt{\dfrac 13}$