The self-organized criticality in Ehrenfest's historical dog-flea model is analyzed by simulating the underlying stochastic process. The fluctuations around the thermal equilibrium in the model are treated as avalanches. We show that the distributions for the fluctuation length differences at subsequent time steps are in the shape of a q-Gaussian (the distribution which is obtained naturally in the context of nonextensive statistical mechanics) if one avoids the finite-size effects by increasing the system size. We provide clear numerical evidence that the relation between the exponent tau of avalanche size distribution obtained by maximum-likelihood estimation and the q value of appropriate q-Gaussian obeys the analytical result recently introduced by Caruso [Phys. Rev. E 75, 055101(R) (2007)]. This allows us to determine the value of q-parameter a priori from one of the well-known exponents of such dynamical systems.