Despite the quantum nature of molecules, classical mechanics is often employed to describe molecular motions that play a fundamental role in a wide range of phenomena including chemical reactions. This is due to the need of assigning well-defined positions to the atomic nuclei during the time evolution of the system in order to describe unambiguously the molecular motions, whereas quantum mechanics provides information on probabilistic nature only. One would like to employ a quantum molecular trajectory that defines rigorously the instantaneous nuclear positions and, simultaneously, guarantees the conservation of all quantum mechanics predictions unlike the classical trajectory. We argue that such a quantum molecular trajectory can be formally defined and we prove that it corresponds to a single Bohm trajectory. Our analysis establishes a clear correspondence between the statistical properties of the trajectory and the quantum expectation values. The obvious and undeniable benefit is that of dealing with a quantum methodology fully characterizing the molecular motions without any reference to classical mechanics.