  The generic quantum $\tau_2$-model (also known as Baxter-Bazhanov-Stroganov
(BBS) model) with periodic boundary condition is studied via the off-diagonal
Bethe Ansatz method. The eigenvalues of the corresponding transfer matrix
(solutions of the recursive functional relations in $\tau_j$-hierarchy) with
generic site-dependent inhomogeneity parameters are given in terms of an
inhomogeneous T-Q relation with polynomial Q-functions. The associated Bethe
Ansatz equations are obtained. Numerical solutions of the Bethe Ansatz
equations for small number of sites indicate that the inhomogeneous T-Q
relation does indeed give the complete spectrum.
