  We generalize the Fej\'er-Riesz operator systems defined for the circle group
by Connes and van Suijlekom to the setting of compact matrix quantum groups and
their ergodic actions on C*-algebras. These truncations form filtrations of the
containing C*-algebra. We show that when they and the containing C*-algebra are
equipped with suitable quantum metrics, then under suitable conditions they
converge to the containing C*-algebra for quantum Gromov-Hausdorff distance.
Among other examples, our results are applicable to the quantum groups
$SU_q(2)$ and their homogeneous spaces $S^2_q$.
