  The off-diagonal Bethe ansatz method is generalized to the integrable model
associated with the $sp(4)$ (or $C_2$) Lie algebra. By using the fusion
technique, we obtain the complete operator product identities among the fused
transfer matrices. These relations, together with some asymptotic behaviors and
values of the transfer matrices at certain points, enable us to determine the
eigenvalues of the transfer matrices completely. For the periodic boundary
condition case, we recover the same $T-Q$ relations obtained via conventional
Bethe ansatz methods previously, while for the off-diagonal boundary condition
case, the eigenvalues are given in terms of inhomogeneous $T-Q$ relations,
which could not be obtained by the conventional Bethe ansatz methods. The
method developed in this paper can be directly generalized to generic $sp(2n)$
(i.e., $C_n$) integrable model.
