The use of traditional optimization methods in engineering design problems, specifically in aerodynamic and infrared stealth optimization for engine nozzles, requires a large number of objective function evaluations, therefore introducing a considerable challenge in terms of time constraints. In this paper, this limitation is addressed by using a sample-efficient multi-objective Bayesian optimization that takes Kriging as a surrogate model and Expected Hypervolume Improvement as the infill criterion. Using this approach, the probabilistic model is continuously established and updated, and the approximate Pareto front is obtained at a relatively small computational budget. The objective of this work is to evaluate the applicability of employing a multi-objective Bayesian optimization framework for the aerodynamic-infrared shape optimization of an elliptical double serpentine nozzle at 6 km flight condition, where the objective functions are evaluated by means of high-fidelity computational fluid dynamics and reversed Monte Carlo ray tracing simulations. We achieve good results in both infrared radiation signature reduction and aerodynamic performance improvement with a reasonable number of evaluations, indicating that the proposed method is effective and efficient for tackling the computationally intensive optimization challenges in the aircraft design.