Mathematical model of gravitational wave detector is described by the Hamiltonian of the quantum oscillator perturbed by a periodic potential, and it's evolution is described by the Wigner function $W(p,q, t)$. We prove that there exists a stochastic representation of $W(p,q, t)$ as the mathematical expectation of a functional of the Poisson random jump process in the phase space. The stochastic representation of the Wigner function is used for the Monte- Carlo simulation of the quantum evolution of the measuring device and the output signal of a realistic gravitational wave detector like LIGO gravitometer. The main conclusion is that the quantum of interaction between the gravitational wave and the detector reduces the confidence of measurements.