Soliton molecules, a frequently observed phenomenon in most mode-locked lasers, have intriguing characteristics comparable to their matter molecule counterparts. However, there are rare explorations of the deterministic control of the underlying physics within soliton molecules. Here, we demonstrate the bistable response of intramolecular motion to external stimuli and identify a general approach to excite their quasi-periodic oscillations. By introducing frequency-swept gain modulation, the intrinsic resonance frequency of the soliton molecule is observed in the simulation model. Applying stronger modulation, the soliton molecule exhibits divergent response susceptibility to up- and down-sweeping, accompanied by a jump phenomenon. Quasi-periodic intramolecular oscillations appear at the redshifted resonance frequency. Given the leading role of bistability and quasi-periodic dynamics in nonlinear physics, our research provides insights into the complex nonlinear dynamics within dissipative soliton molecules. It may pave the way to related experimental studies on synchronization and chaos at an ultrafast time scale.