A recently suggested equation of state with the induced surface tension is generalized to the case of quantum gases with mean-field interaction. The self-consistency conditions of such a model and the necessary one to obey the Third Law of thermodynamics are found. The quantum virial expansion of the Van der Waals models of such a type is analyzed and its virial coefficients are given. In contrast to traditional beliefs, it is shown that an inclusion of the third and higher virial coefficients of the gas of hard spheres into the interaction pressure of the Van der Waals models either breaks down the Third Law of thermodynamics or does not allow one to go beyond the Van der Waals approximation at low temperatures. It is demonstrated that the generalized equation of state with the induced surface tension allows one to avoid such problems and to safely go beyond the Van der Waals approximation. Besides, the effective virial expansion for quantum version of the induced surface tension equation of state is established and all corresponding virial coefficients are found exactly.The explicit expressions for the true quantum virial coefficients of an arbitrary order of this equation of state are given in the low density approximation. A few basic constraints on such models which are necessary to describe the nuclear and hadronic matter properties are discussed.