Theoretical studies are reported for the quantum dynamical and statistical properties of an atom laser. The model is based on the generalized deformed oscillator algebra in which the field radiation operators are deformed by an operator-valued function f(N) of the photon number N. The Hamiltonian of the system is solved analytically under the rotating wave approximation. In particular, it is shown that the nonlinearity introduced by q-deformation leads to quadrature squeezing effects and a sub-Poisson distribution for the atom laser beam.