The furnished article primely focuses on the second-grade ternary hybrid flow and thermal characteristics triggered by vertical stretching sheets, featured with buoyant and thermo-convection effect subjects to Thomson and Troian velocity slip at the boundary. The ternary hybrid Nanofluid is composed of engine oil blended with three distinct nanocomposites: Ag, Cu, and graphene. Furthermore, the thermal boundary layer equation addresses the nonlinear exponential heat source and the thermo-convection effect. The aforementioned assumption leads to a schematic model framed with coupled partial differential equations revamped into an array of coupled ordinary differential equations via some relevant similarity variable, which are numerically solved by the bvp4c method. The novel contribution embraces the impact of Thomson and troian slip velocity, thermo-convection parameter, and exponential power law index alongside the second-grade fluid factor and suction parameter over the flow and thermal trajectory outline. It is noteworthy that the augmented variation in Thomson and Troian slip velocity and suction parameters diminished the flow pattern, but the opposite trend is noted with the escalating parameters such as second-grade fluid factor and critical shear rate. Moreover, the thermal diffusion profile appears in an elevated pattern with increased Thomson and Troian slip velocity, exponential heat source, and power law index. Beyond these, the heat transfer rate in the vicinity of the wall promptly rises with the augmentation in the Second-grade fluid parameter, critical Shear rate, thermos convection parameter, suction parameter, and power law index. Researchers can build on these findings to explore new material combinations and refine theoretical fluid dynamics and thermal management models.