In this paper, we study the nonlinear $p$-Laplacian equation $$-Δ_{p} u+V(x)|u|^{p-2}u=f(x,u) $$ with positive and periodic potential $V$ on the lattice graph $\mathbb{Z}^{N}$, where $Δ_{p}$ is the discrete $p$-Laplacian, $p \in (1,\infty)$. The nonlinearity $f$ is also periodic in $x$ and satisfies the growth condition $|f(x,u)| \leq a(1+|u|^{q-1})$ for some $ q>p$. We first prove the equivalence of three function spaces on $\mathbb{Z}^{N}$, which is quite different from the continuous case and allows us to remove the restriction $q>p^{*}$ in [SW10], where $p^{*}$ is the critical exponent for $ W^{1,p}(Ω) \hookrightarrow L^{q}(Ω)$ with $Ω\subset \mathbb{R}^{N}$ bounded. Then, using the method of Nehari [Neh60, Neh61], we prove the existence of ground state solutions to the above equation.