Green colour emitting graphene quantum dots (GQDs) are prepared by a simple acid reflux reaction of graphene oxide (GO) produced using a modified Hummer's method. Structural and morphological characterizations of such GQDs are performed using spectroscopic (FTIR, UV-vis and photoluminescence) and microscopic (transmission electron microscopy) techniques. These studies reveal the formation of stable, uniform spherical particles of GQDs which emit a green colour and possess surface functional moieties such as epoxide, hydroxyl (-OH) and carboxyl (-COOH) groups. Further, the possibility of immobilizing biomolecules on GQDs using these surface active functional groups is explored. As an example, an enzyme namely horseradish peroxidase (HRP) is shown to be anchored on these GQDs using a coupling reaction between an acid and amine leading to the formation of a peptide amide bond. Enzymatic activity of HRP is investigated by simply drop-casting HRP-immobilized GQDs onto a glassy carbon electrode. Electrochemical studies clearly reveal the formation of a well-defined redox peak and the dependence of redox peak current on scan rate suggests that the HRP enzyme is anchored onto the electrode, surface confined and exhibits a direct electron transfer process that is predominantly controlled by a diffusion process. These HRP-functionalized GQDs are used as a sensing platform for hydrogen peroxide detection. This particular electrochemical biosensor shows the sensitivity values of 0.905 and 7.057 mu A/mM and detection limits of similar to 530 nM and 2.16 mu M along with a fast response time of similar to 2-3 s.