•The multi-objective optimization method is developed for thermal-fluid problems.•Pareto-optimal solutions based on the multi-objective function are provided.•A novel model for heat transfer enhancement and pressure drop reduction.•The influence of non-Newtonian fluid on the microchannel heat transfer system is considered. This paper presents a multi-objective topology optimization method for convective heat transfer problems based on Navier-Stokes and heat transport equations. In this research, the pressure drop (or energy dissipation) function and the recoverable thermal power function are combined and optimized as a multi-objective function. A Pareto algorithm is constructed, based on a weighted-sum method using different weight coefficients. The Pareto solutions composed of a set of optimal solutions are obtained to reveal the trade-off relationship between the objective functions. The effects of non-Newtonian fluid on the flow-channel arrangement and heat transfer performance are numerically studied. In the numerical method, the interpolation term in the density-based topology optimization method is modified, while the filtering method is used to solve the problems of grayscale and the flow channel discontinuity. The results show that the differences between the optimal structures for non-Newtonian fluid and Newtonian fluid are more obvious at Re = 0.01 in the dual-terminal devices. Under the same situation, blood (non-Newtonian fluid) has greater energy dissipation and better heat transfer performance than water (Newtonian fluid).