  The nested off-diagonal Bethe ansatz is generalized to study the quantum spin
chain associated with the $SU_q(3)$ R-matrix and generic integrable
non-diagonal boundary conditions. By using the fusion technique, certain closed
operator identities among the fused transfer matrices at the inhomogeneous
points are derived. The corresponding asymptotic behaviors of the transfer
matrices and their values at some special points are given in detail. Based on
the functional analysis, a nested inhomogeneous T-Q relations and Bethe ansatz
equations of the system are obtained. These results can be naturally
generalized to cases related to the $SU_q(n)$ algebra.
