We report on a small and simple graphene-based potentiometric sensor for the measurement of intracellular glucose concentration. A fine borosilicate glass capillary coated with graphene and subsequently immobilized with glucose oxidase (GOD) enzyme is inserted into the intracellular environment of a single human cell. The functional groups on the edge plane of graphene assist the attachment with the free amine terminals of GOD enzyme, resulting in a better immobilization. The sensor exhibits a glucose-dependent electrochemical potential against an Ag/AgCl reference microelectrode which is linear across the whole concentration range of interest (10 - 1000 μM). Glucose concentration in human fat cell measured by our graphene-based sensor is in good agreement with nuclear magnetic resonance (NMR) spectroscopy.