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This paper is concerned with the following periodic Hamiltonian elliptic system $\{-\Delta \varphi +V\left(x\right)\varphi =G_{\psi }(x,\varphi ,\psi )\text{in}{ℝ}^N,-\Delta \psi +V\left(x\right)\psi =G_{\varphi }(x,\varphi ,\psi )\text{in}{ℝ}^N,\varphi \left(x\right)\to 0\text{and}\psi \left(x\right)\to 0\text{as}|x|\to \infty .$ Assuming the potential V is periodic and 0 lies in a gap of $\sigma (-\Delta +V)$, $G(x,\eta )$ is periodic in x and asymptotically quadratic in $\eta =(\varphi ,\psi )$, existence and multiplicity of solutions are obtained via variational approach.