We use exact diagonalization to study quantum chaos in a simple model with two bosonic and one fermionic degree of freedom. Our model has a structure similar to the BFSS matrix model (compactified supersymmetric Yang-Mills theory), and is known to have a continuous energy spectrum. To diagnose quantum chaos, we consider energy level statistics and the out-of-time-order correlators (OTOCs). We find that OTOCs exhibit monotonous growth down to the lowest temperatures, thus indicating Lyapunov instability at all temperatures. This is in contrast to the purely bosonic models like pure Yang-Mills theory, which are non-chaotic at low temperatures because of their gapped energy spectrum. Despite the apparently chaotic behavior at all temperatures, we find that the energy level statistics undergoes a sharp transition between non-chaotic, one-dimensional low-energy states and delocalized high-energy states with random-matrix-type statistics of energy levels.