The objective of this paper is to analyze the effect of hydraulic diameter and channel shape on the thermal and hydrodynamic characteristics of a microchannel cooled by Graphene–Platinum/water hybrid nanofluid for electronic cooling applications. The study was performed numerically using mathematical software called Maple 19.0. Microchannels having square and rectangular cross-sections, and hydraulic diameters ranging from 200 µm to 1,000 µm were taken into consideration. Thermal resistance, heat transfer coefficient, pressure drop, and friction factor were evaluated for different conditions and their corresponding graphs are presented and discussed. It was evident from the results that low thermal resistance and high heat transfer coefficient was achieved upon decreasing the hydraulic diameter, which is favorable for the cooling of electronic chips and devices. Based on the Reynolds number, the heat transfer coefficient increased by 2–4 times for both rectangular and square microchannels, on decreasing the hydraulic diameter from highest value (1,000 µm) to lowest value (200 µm). However, friction factor and pressure drop increased for channels with lower hydraulic diameters. In addition, rectangular microchannels exhibited better heat transfer performance, while square microchannels had lower friction factor and pressure drop. Rectangular microchannels presented a maximum enhancement of 30% in heat transfer coefficient and a reduction of 18% in thermal resistance, when compared to square microchannels. The results also suggested that the performance of microchannels with 500 µm hydraulic diameter is balanced, considering both heat transfer performance and pressure drop constraints.