The effect of copper oxide on migration and interaction of protons in barium zirconate was investigated using density functional theory. One copper atom was substituted for a zirconium atom site, and a proton was added to a $3{ imes}3{ imes}3$ barium zirconate superstructure. An energy barrier of 0.89 eV for proton migration was the highest among several energy barriers. To investigate the interaction between multiple protons and a copper atom, two protons were added to the superstructure. Various proton positions were determined by the interaction between the two protons and the copper atom.