The search for a consistent theory of quantum gravity has motivated the development of radically different approaches. This seeks consists of constructing a mathematical apparatus that encapsulates both concepts of quantum theory and general relativity. However, none approach has been definitive and the problem remains open. As the quantization of the metric is an alternative, this paper shows how a metric operator may be explicitly obtained by introducing a temporal operator, defining an induced metric and invoking some spacetime symmetries. This makes it possible to relate the effective acoustic metric to the model proposed here. The metric operator equations are expressed in terms of a hamiltonian operator describing the degrees of freedom of quantum vaccum whose dynamics gives rise to the metric field. These findings may help understand and study the quantum vacuum at Planck scale, consisting of one more tool for the community working on quantization of gravity.