Dynamical Casimir effect (DCE) designates a plethora of phenomena characterized by generation of photons (or quanta of some other field) from vacuum due to time-dependent variations of the geometry (dimensions) or material properties (e.g., the dielectric constant or conductivity) of some macroscopic system. The circuit QED architecture is a handy platform for the implementation of DCE and its analogues, since both the cavity's and artificial atoms' properties can be rapidly modulated by external magnetic field. In particular, the generation of photons and light-matter entangled states from the initial vacuum state can take place for a resonant time-modulation of the atomic transition frequency or the atom-field coupling strength, when the atom is directly coupled to the field via the dipole interaction (as described by the Quantum Rabi Model). In this case, one can think of the atom as a microscopic constituent of the intracavity medium. Here we propose a scheme for the generation of photons from vacuum due to the time-modulation of some quantum system that is "invisible" to the field, namely, some artificial 2-level atom (nonstationary qubit) that is indirectly coupled to the field via some ancilla quantum subsystem. We consider the simplest case when the ancilla is a stationary qubit, coupled via the dipole interaction both to the cavity field and the nonstationary qubit, but our results can be easily generalized to more complex atoms. We find that a small number of photons can be generated from the system ground state under resonant modulations even when the nonstationary qubit is far detuned from both the ancilla and the cavity. Moreover, for specific relations between the natural frequencies of the three subsystems the photon generation rate can be strongly enhanced due to the anti-crossings in the energy levels of the bare Hamiltonian, when the system is driven to a triparite entangled state. We attest our approximate analytic results by numeric simulations and show that the photon generation from vacuum persists even in the presence of common dissipation mechanisms. Thus, our scheme could find applications in the engineering of effective interactions and entangled states in hybrid cavity QED systems, when the nonstationary qubit is designed to undergo fast large-amplitude modulations, while the cavity field is coupled to another stationary atom.