We analyze the noise in liquid-gated, room temperature, graphene quantum dots. These devices display extremely large noise amplitudes. The observed noise is explained in terms of a charge noise model by considering fluctuations in the applied source-drain and gate potentials. We show that the liquid environment and substrate have little effect on the observed noise and as such attribute the noise to charge trapping/detrapping at the disordered graphene edges. The trapping/detrapping of individual charges can be tuned by gating the device, which can result in stable two-level fluctuations in the measured current. These results have important implications for the use of electronic graphene nanodevices in single-molecule biosensing.