Graphene quantum dots (GQDs) have become increasingly important for applications in energy materials, optical devices and biosensors. Here we report a facile technique to fabricate a nanofibrous membrane of GQDs by electrospinning water-soluble GQDs with polyvinyl alcohol (PVA) directly. The structure and fluorescence properties of the fabricated PVA/GQD nanofibrous membrane were investigated using scanning and transmission electron microscopy, and fluorescence microscopy. It was found that the electrospun PVA/GQD nanofibrous membrane has a three-dimensional structure with a high surface area to volume ratio, which is beneficial for the adsorption of electrolytes and the diffusion of reactants. For the first time, the created PVA/GQD nanofibrous membrane was utilized to fabricate dual-purpose fluorescent and electrochemical biosensors for highly sensitive determination of hydrogen peroxide (H2O2) and glucose. The experimental results indicated that the fluorescence intensity of the nanofibrous membrane decreased linearly with increasing H2O2 concentration, because the addition of H2O2 leads to fluorescence quenching of the GQDs, which endows the fabricated nanofibrous membrane with fluorescence activity. Besides, after binding glucose oxidase onto the created nanofibrous membrane, the fabricated nanofibrous membrane showed high sensitivity and selectivity for glucose detection. In addition, the PVA/GQD nanofibrous membrane can also be directly electrospun onto an electrode for electrochemical detection of H2O2. This novel nanofibrous membrane exhibits excellent catalytic performance and fluorescence activity, and therefore has potential applications for the highly stable, sensitive, and selective detection of H2O2 and glucose.