Microelectronic technologies are progressing rapidly. As devices shrink in size, they produce a substantial heat flux that can adversely affect performance and shorten their lifespan. Conventional cooling methods, such as forced-air heat transfer and essential heat sinks, are inadequate for managing the elevated heat flux generated by these devices. Consequently, microchannel heat sinks have been developed to address this challenge. The present research is intended to study forced flow convection and heat transfer in a cone–column combined microchannel heat sink (MCHS). This study examines a regularly shaped MCHS to evaluate its heat transfer rate. The heat transfer medium employed is a graphene–water nanofluid, and the heat sink’s base is assumed to maintain a constant heat flux. The Galerkin weighted finite element method solves the nanofluid’s governing partial differential equations. This thesis investigates the impact of varying intake velocities on the Reynolds number (100 ≤ Re ≤ 900), externally applied heat flux (104 ≤ q ≤ 106), and the volumetric ratio of nanoparticles (0.001 ≤ φ ≤ 0.04). The study conducts a mathematical analysis to explore how these parameters affect pressure drop, friction factor, average Nusselt number, average substrate temperature, and heat transfer enhancement. The findings are compared with those of a conventional MCHS as the Re increases. The results are analyzed and visually represented through isothermal lines for temperature contours and streamlines for velocity. An increase in the inlet velocity of the water–graphene nanofluid significantly enhances heat transfer and thermal efficiency, achieving improvements of approximately 27.00% and 21.21%, respectively. The research demonstrates that utilizing water–G as a smart coolant with the cone–column combined MCHS enhances thermal efficiency by 4.05% compared to standard water. A comparison of the hydraulic performance index at the substrate reveals that the cone–column combined MCHS is significantly more effective at dissipating heat than the traditional MCHS.