A facile bottom-up method for the synthesis of highly fluorescent graphene quantum dots (GQDs) has been developed using a one-step pyrolysis of a natural amino acid, L-glutamic acid, with the assistance of a simple heating mantle device. The developed GQDs showed strong blue, green and red luminescence under irradiation with ultra-violet, blue and green light, respectively. Moreover, the GQDs emitted near-infrared (NIR) fluorescence in the range 800–850 nm with an excitation-dependent manner. This NIR fluorescence has a large Stokes shift of 455 nm, providing a significant advantage for the sensitive determination and imaging of biological targets. The fluorescence properties of the GQDs, such as the quantum yields, fluorescence life times, and photostability, were measured and the fluorescence quantum yield was as high as 54.5%. The morphology and composites of the GQDs were characterized using TEM, SEM, EDS, and FT-IR. The feasibility of using the GQDs as a fluorescent biomarker was investigated through in vitro and in vivo fluorescence imaging. The results showed that the GQDs could be a promising candidate for bioimaging. Most importantly, compared to the traditional quantum dots (QDs), the GQDs are chemically inert. Thus, the potential toxicity of the intrinsic heavy metal in the traditional QDs would not be a concern for GQDs. In addition, the GQDs possessed an intrinsic peroxidase-like catalytic activity that was similar to graphene sheets and carbon nanotubes. Coupled with 2,2′-azino-bis(3-ethylbenzothiazoline-6-sulphonic acid) (ABTS), the GQDs can be used for the sensitive detection of hydrogen peroxide with a limit of detection of 20 μM.