Your hyperboloid is unbounded:
RegionMeasure[
ImplicitRegion[(x^2/A^2 + y^2/A^2 - z^2/C^2 <= 1) /. {A -> 2,
C -> 6}, {x, y, z}]]
As for the integral, I would rather avoid the underscript and other 2D notations:
Integrate[{{y^2 + z^2, -x y, -x z},
{-y x, x^2 + z^2, -y z},
{-z x, -z y, x^2 + y^2}},
Element[{x, y, z}, Cuboid[]]] // MatrixForm