The quantum breathers (QB) state in terms of two-phonon bound state (TPBS) via detailed quantum calculations was already shown in ferroelectrics in a periodic boundary condition. In this work on metamaterials for application in antenna arrays as split-ring resonators (SRR), only the temporal variation of the number of quanta in a non-periodic boundary condition is detailed for a Klein-Gordon lattice. A generalized Hamiltonian containing dielectric permittivity is considered which was then quantized by bosonic operators to study the temporal evolution of quanta. Moreover, the time of redistribution of quanta that is proportional to QB's lifetime in femtosecond shows significant variation with different quanta at different values of dielectric permittivity, which has implications in the applications in relevant devices.