Resonant dielectric nanostructures represent a promising platform for light manipulation at the nanoscale. In this paper, we describe an active photonic system based on Ge(Si) quantum dots coupled to silicon nanodisks. We show that Mie resonances govern the enhancement of the photoluminescent signal from embedded quantum dots due to a good spatial overlap of the emitter position with the electric field of Mie modes. We identify the coupling mechanism, which allows for engineering the resonant Mie modes through the interaction of several nanodisks. In particular, the mode hybridization in a nanodisk trimer results in an up to 10-fold enhancement of the luminescent signal due to the excitation of resonant antisymmetric magnetic and electric dipole modes.