Return distributions of the coherent noise model are studied for the system-size-independent case. It is shown that, in this case, these distributions are in the shape of q Gaussians, which are the standard distributions obtained in nonextensive statistical mechanics. Moreover, an exact relation connecting the exponent tau of avalanche size distribution and the q value of appropriate q Gaussian has been obtained as q=(tau+2)/tau. Making use of this relation one can easily determine q parameter values of the appropriate q Gaussians a priori from one of the well-known exponents of the system. Since the coherent noise model has the advantage of producing different tau values by varying a model parameter sigma, clear numerical evidences on the validity of the proposed relation have been achieved for various cases. Finally, the effect of the system size has also been analyzed and an analytical expression has been proposed, which is corroborated by the numerical results.