Carbon-based nanosheet membranes with functionalized pores have great potential as water treatment membranes. In this study, the separation of Hg2+ and Cu2+ as heavy metal ions from aqueous solutions using a functionalized γ-graphyne-1 nanosheet membrane is investigated by molecular dynamics simulations. The simulation systems consist of a γ-graphyne-1 nanosheet with -COOH or -NH2 functional groups on the edge of pores placed in an aqueous solution containing CuCl2 and HgCl2. An external electric field is applied as a driving force across the membrane for the separation of heavy metal ions using these functionalized pores. The ion-membrane and water molecule-membrane interaction energies, the radial distribution function of cations, the retention time and permeation of ions through the membrane, the density profile of water and ions, and the hydrogen bond in the system are investigated, and these results reveal that the performance of -NH2-functionalized γ-graphyne-1 is better than that of -COOH-functionalized γ-graphyne-1 in the separation of Cu2+, while the Hg2+ cations encounter a high energy barrier as they pass through the membrane, especially in the -COOH-functionalized pore, due to their larger ionic radius and the smaller pore size of this membrane.