[en] In this paper, an optomechanical cavity that is quadratically coupled to the cavity field and formed by a micropillar with Bragg reflectors and a thin dielectric membrane, is studied. In addition, it is considered that this interaction occurs in the presence of Kerr medium, external laser field and damping effects (for cavity field and moving Bragg reflector). Using the Heisenberg-Langevin approach, the dynamics of quantum correlation functions (for cavity field and moveable reflector) and squeezing parameters for quadratures of the cavity field are investigated. In each case, the influences of optomechanical coupling, detuning parameter, thermal mean number of cavity photons and phonons, damping parameters and Kerr medium on the above nonclassicality features are analyzed in detail. It is illustrated that the amount of the above-mentioned physical phenomena can be controlled by appropriately choosing the evolved parameters. Also, we show that photon and phonon blockades emerge in some special cases.