By using canonical quantization method of damped harmonic oscillator, the quantum of dissipative mesoscopic RLC series and parallel circuit was realized. And on this basis, the quantum fluctuations of charge and inductance flux linkage as the same as voltage and current in circuit under vacuum state were also studied through the methods. The results show that charge and inductance flux linkage as the same as voltage and current in circuit under vacuum state respectively exist quantum fluctuations. And the size of the quantum fluctuations and quantum fluctuations plot all depend on device parameters in the circuit, which decaying exponentially with time.