A generalization of the quantum van der Waals equation of state for a multi-component system in the grand canonical ensemble is proposed. The model includes quantum statistical effects and allows to specify the parameters characterizing repulsive and attractive forces for each pair of particle species. The model is straightforwardly applied to the description of asymmetric nuclear matter and also for mixtures of interacting nucleons and nuclei. Applications of the model to the equation of state of an interacting hadron resonance gas are discussed.