Graphene and graphene oxide (G-O) have been demonstrated to be excellent filters for various gases and liquids, showing potential applications in areas such as molecular sieving and water desalination. In this paper, the selective trans-membrane transport properties of alkali and alkaline earth cations through a membrane composed of stacked and overlapped G-O sheets ("G-O membrane") are investigated. The thermodynamics of the ion transport process reveal that the competition between the generated thermal motions and the interactions of cations with the G-O sheets results in the different penetration behaviors to temperature variations for the considered cations (K(+), Mg(2+), Ca(2+), and Ba(2+)). The interactions between the studied metal atoms and graphene are quantified by first-principles calculations based on the plane-wave-basis-set density functional theory (DFT) approach. The mechanism of the selective ion trans-membrane transportation is discussed further and found to be consistent with the concept of cation-π interactions involved in biological systems. The balance between cation-π interactions of the cations considered with the sp(2) clusters of G-O membranes and the desolvation effect of the ions is responsible for the selectivity of G-O membranes toward the penetration of different ions. These results help us better understand the ion transport process through G-O membranes, from which the possibility of modeling the ion transport behavior of cellular membrane using G-O can be discussed further. The selectivity toward different ions also makes G-O membrane a promising candidate in areas of membrane separations.