We study a (quenched) random-field quantum model of an anharmonic crystal for displacive structural phase transitions in spherical approximation: the random-field quantum spherical (ferroelectric) model. For stationary ergodic random fields its behavior depends on the quantum parameter of the model and on the expectation and covariance of the field. If quantum fluctuations are small enough not to destroy the phase transition, then it can be suppressed when the field fluctuations are large. For the field of independent identically distributed random variables and the short-range interaction we obtain that the lower critical dimensionality dl=4 (dl=2 for the zero-field) and that it decreases for long-range interactions.