In this paper, by considering a system consisting of a single two-level trapped ion interacting with a single-mode quantized radiation field inside a lossless cavity, the temporal evolution of the ionic and the cavity-field quantum statistical properties including photon-counting statistics, quantum fluctuations of the field quadratures and quantum fluctuations of the ionic dipole variables are investigated. It is found that in the Lamb-Dicke limit it is possible for the cavity-field to evolve into a nonclassical state (squeezed state or state with sub-Poissonian statistics) and this possibility is solely affected by the internal and external quantum dynamics of the ion. Also it is found that the dipole squeezing may occur in the dynamics of the trapped ion which sensitively depends on quantum statistics of the cavity-field.