The present study investigates the catalytic performance of a hg−C3N4 quantum dot aimed at enhancing electrochemical water splitting, using the first-principles density functional theory. The size of the considered quantum dot lies within the range reported experimentally (2nm - 4nm) [Zhou et al. ACS Nano 9, 12480 (2015)]. The nitrogen vacancies are created in the considered hg−C3N4 structure to simulate the realistic scenario, as the presence of nitrogen and carbon defects are reported in the synthesized hg−C3N4 quantum dots. First, the structural and vibrational properties are computed to ensure the stability of the nitrogen deficient hg−C3N4 quantum dots, and subsequently, their electronic and hydrogen evolution reaction (HER) properties are investigated. The calculate HER parameters, i.e., adsorption energies, Gibbs free energies, and overpotentials demonstrate that the considered hg−C3N4 quantum dot with nitrogen vacancies can be used as a moderately effective electrocatalyst for HER performance. We also considered the quantum dot to be dissolved in water and ethanol, and find that the overpotential gets drastically reduced to 16 mV for the alcohol dissolved quantum dot, while some significant reduction is seen for the aqueous solution also. As a result, this study suggests that the nitrogen-deficient hg−C3N4 quantum dots dissolved in ethanol are excellent candidates for catalysis aimed at sustainable hydrogen production via electrochemical water splitting.