  We study the discrete spectrum of the two-particle Schr\"odinger operator
$\hat H_{\mu\lambda}(K),$ $K\in\mathbb{T}^2,$ associated to the Bose-Hubbard
Hamiltonian $\hat {\mathbb H}_{\mu\lambda}$ of a system of two identical bosons
interacting on site and nearest-neighbor sites in the two dimensional lattice
$\mathbb{Z}^2$ with interaction magnitudes $\mu\in\mathbb{R}$ and
$\lambda\in\mathbb{R},$ respectively. We completely describe the spectrum of
$\hat H_{\mu\lambda}(0)$ and establish the optimal lower bound for the number
of eigenvalues of $\hat H_{\mu\lambda}(K)$ outside its essential spectrum for
all values of $K\in\mathbb{T}^2.$ Namely, we partition the
$(\mu,\lambda)$-plane such that in each connected component of the partition
the number of bound states of $\hat H_{\mu\lambda}(K)$ below or above its
essential spectrum cannot be less than the corresponding number of bound states
of $\hat H_{\mu\lambda}(0)$ below or above its essential spectrum.
