The commuting graph of an arbitrary ring R, denoted by ${Gamma}(R)$, is a graph whose vertices are all non-central elements of R, and two distinct vertices a and b are adjacent if and only if ab = ba. In this paper, we investigate the connectivity, the diameter, the maximum degree and the minimum degree of the commuting graph of group ring $Z_nQ_8$. The main result is that $Gamma(Z_nQ_8)$ is connected if and only if n is not a prime. If $Gamma(Z_nQ_8)$ is connected, then diam($Z_nQ_8$)= 3, while $Gamma(Z_nQ_8)$ is disconnected then every connected component of $Gamma(Z_nQ_8)$ must be a complete graph with a same size. Further, we obtain the degree of every vertex in $Gamma(Z_nQ_8)$, the maximum degree and the minimum degree of $Gamma(Z_nQ_8)$.