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Density functional theory (DFT) can be quite advantageous in advancing the field of catalysis because of the microscopic insights it provides, and thus can guide experimental searches of novel catalysts. Several recent works have demonstrated that low-dimensional materials can be very efficient catalysts. Graphene quantum dots (GQDs) have gained much attention in past years due to their unique properties like low toxicity, chemical inertness, biocompatibility, crystallinity, etc. These properties of GQDs which are due to quantum confinement and edge effects facilitate their applications in various fields like sensing, photoelectronics, catalysis, and many more. Furthermore, the properties of GQDs can be enhanced by doping and functionalization. In order to understand the effects of functionalization by oxygen and boron based groups on the catalytic properties relevant to the hydrogen-evolution reaction (HER), we perform a systematic study of GQDs functionalized with the oxygen (O), borinic acid (BC$_2$O), and boronic acid (BCO$_2$ ). All calculations that included geometry optimization, electronic and adsorption mechanism, were carried out using the Gaussian16 package, employing the hybrid functional B3LYP, and the basis set 6-31G(d,p). With the variation in functionalization groups in GQDs, we observe significant changes in their electronic properties. The adsorption energy E$_{ads}$ of hydrogen over O-GQD, BC$_2$O-GQD, and BCO$_2$-GQD is -0.059 eV, -0.031 eV and -0.032 eV respectively. Accordingly, Gibbs free energy ($ΔG$) of hydrogen adsorption is extraordinarily near the ideal value (0 eV) for all the three types of functionalized GQDs. Thus, the present work suggests pathways for experimental realization of low-cost and multifunctional GQDs based catalysts for clean and renewable hydrogen energy production.