The vibrational relaxation rate of a "quantum" molecule embedded in a "quantum" host is approached from the perspective of a short-time expansion of the quantum force-force time correlation function. A general path integral Monte Carlo scheme is developed to compute the coefficients of the expansion, which involve the quantum thermal average of operators that depend on both position and momentum. A simple ansatz is used to connect the short-time and long-time behavior of the force-force correlation function. The method is tested on a simple model problem. Improvements of the approach and further applications are discussed.