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The dynamical properties of an overdamped Brownian particle moving in an asymmetric bistable system with quantum fluctuations are investigated. Within the strong-friction limit (the quantum Smoluchowski regime), the analytic expression for the relaxation time of the system is derived by means of the projection-operator method, in which the effects of the memory kernels are taken into account. Based on the relaxation time, we consider both the overdamped quantum case and its classical counterpart.In these contexts, the effects of the quantum fluctuations and the asymmetry of the potential are discussed. It is found that: (i) The quantum effects in an asymmetric bistable system on time scales of the relaxation process are more prominent for lower temperatures and smaller asymmetries of the potential. (ii) The quantum effects speed up the rate of fluctuation decay of the state-space variable for lower temperatures. (iii) The asymmetry of the potential first slows down the rate of fluctuation decay of the state-space variable and then increases it.
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