We present the first multi-fidelity Bayesian optimization approach for solving inverse problems in the quantum control of prototypical quantum systems. Our approach automatically constructs time-dependent control fields that enable transitions between initial and desired final quantum states. Most importantly, our Bayesian optimization approach gives impressive performance in constructing time-dependent control fields, even for cases that are difficult to converge with existing gradient-based approaches. We provide detailed descriptions of our machine learning methods as well as performance metrics for a variety of machine learning algorithms. Taken together, our results demonstrate that Bayesian optimization is a promising approach to efficiently and autonomously design control fields in general quantum dynamical systems.