  In this paper, we study the exact solution of the $q$-deformed $D^{(1)}_3$
quantum lattice model with non-diagonal open boundary condition. We demonstrate
the crossing symmetry of the transfer matrix and obtain the quantum
determinant. We construct the independent transfer matrix fusion identities and
show that the fusion processes can be closed. Based on the fusion hierarchies
and polynomial analysis, we obtain the inhomogeneous $T-Q$ relations, exact
energy spectrum and Bethe ansatz equations of the system.
