The thermal superiority of ternary hybrid nanofluids (THNFs) over conventional heat transfer fluid has led to growing interest in their applications. This new type of nanofluid can be customized for cooling systems, heat exchangers, and electronic cooling by carefully selecting nanoparticle types and their volume fraction. Hence, this study seeks to investigate the Heimenz flow in a Williamson THNF over a sheet that stretches or shrinks. The fundamental objective is to assess the effect of the stretching/shrinking parameter, the Weissenberg number, and the nanoparticle volume fraction on the physical quantities and flow profiles. Besides, attention is also given to the occurrences of multiple solutions in this fluid flow situation. By employing a similarity transformation, the governing equations are modified as a simpler form of ordinary differential equations (ODEs). Next, the numerical method is put to use to solve the resulting ODEs system, specifically the bvp4c solver in MATLAB. Significant changes in heat transmission occur due to variations in the Weissenberg number and volume fractions of nanoparticles, particularly when the sheet starts to shrink. The escalating Weissenberg number correlates with growing critical values of the stretching/shrinking parameter, suggesting that both parameters help to hold off the detachment of the boundary layer. These findings emphasize the capacity of THNFs to improve heat transfer performance in numerous applications. This study also reveals that while stretching sheets often have unique solutions, a shrinking sheet has multiple solutions when the shrinking parameter falls within a certain range. By scrutinizing the robustness of these solutions, it was concluded that only one of them maintains stability over an extended period. It is essential to highlight that these present discoveries apply exclusively to the mixture of copper, alumina, and titania. Various mixtures of nanoparticles can demonstrate distinct characteristics of THNFs concerning both flow dynamics and thermal transfer.