Geopolymer concrete uses a geopolymer binder instead of traditional Portland cement; thus, it reduces carbon emissions by a significant amount. In this study, Edge-Oxidized Graphene Oxide (EOGO), a carbon-based nanomaterial, was added into a metakaolin-based geopolymer, and its effect on the mechanical and rheological properties of the mixture was investigated. EOGO was added into the mixture at 0% (control), 0.1%, 0.5%, and 1% of the metakaolin mass. Several experiments were conducted to characterize the properties of the metakaolin–EOGO (MKGO) geopolymer, including its compressive strength, free–free resonance column (FFRC), void content, water absorption, setting time, flow, and rheology. It was found that the compressive strength and stiffness showed their maximum values and the void content was minimized at 0.1% EOGO. In addition, as the EOGO addition rate increased, the setting time tended to shorten, and the fluidity tended to decrease. This suggests that 0.1% EOGO is the most optimal content in metakaolin paste. This study confirms that EOGO is an additive material that can improve the performance of metakaolin-based geopolymers and presents opportunities for the development of sustainable construction materials through optimization of EOGO addition.