The condition on your coefficients is an underdetermined system of quadratic equations. So specializing some of them (for instance setting them $0$), you can try to get a system with a finite number of (complex) solutions, and the chances are good that they are actually rational. Such systems can be handled with SageMath (which uses Singular, msolve or other systems as backend). In the specific case, a possible solution is
\begin{align}
Q_0 &= X^2 - XY + Y^2 + XZ\\\\
Q_1 &= X^2 - XY + YZ\\\\
Q_2 &= X^2 - XZ + YZ + Z^2.
\end{align}