Robot design is a time-consuming process involving repeated experiments in a variety of environments to optimize multiple, possibly conflicting performance metrics. Moreover, the optimal robot performance for a given design depends on how the robot adapts its behavior to its environment. We propose a multi-objective Bilevel Bayesian optimization (MO-BBO) technique to automate the process of form-behavior co-design. The approach expands the Pareto front of multiple metrics by simultaneously exploring the robot design and behavior. MO-BBO uses a bilevel optimization of the acquisition function with design and behavior parameters being the high- and low-level decision variables, respectively. In the low-level, we always choose environment-aware behaviors that maximize each metric. We evaluate MO-BBO in applications to grasping gripper design and bimanual arm placement, and show that our method can efficiently focus samples on the Pareto front and generate a diversity of designs.