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We investigate the quantum chaos of ultracold atoms in a quadruple-well potential, emphasizing the influence of synthetic gauge fields and many-body atomic interactions on the transitions from regular behavior to quantum chaos. We compare the metrics that the eigenvalue statistics of nearest-neighbor spacings distribution and gap ratio analysis, and the eigenfunction statistics using the Kullback–Leibler divergence and the inverse participation ratio, to diagnose the transition from regular to chaotic behavior. Numerically, we obtain the phase diagram of the regular and chaotic behavior on the parameter plane of many-body atomic interactions and synthetic gauge fields. Our results provide insight into the chaotic behavior of ultracold atoms with synthetic gauge fields and have potential applications in the quantum engineering of wave chaos.