The Global Network of Optical Magnetometers for Exotic physics searches (GNOME) is a network of time-synchronized, geographically separated, optically pumped atomic magnetometers that is being used to search for correlated transient signals heralding exotic physics. GNOME is sensitive to exotic couplings of atomic spins to certain classes of dark matter candidates, such as axions. This work presents a data analysis procedure to search for axion dark matter in the form of topological defects: specifically, walls separating domains of discrete degenerate vacua in the axion field. An axion domain wall crossing the Earth creates a distinctive signal pattern in the network that can be distinguished from random noise. The reliability of the analysis procedure and the sensitivity of the GNOME to domain-wall crossings are studied using simulated data.