In this paper, we consider the global well-posedness of solutions to a parabolic-parabolic Keller-Segel model with $p$-Laplace diffusion. We first establish a critical exponent $p^∗=3N/(N+1)$ and prove that when $p> p^∗,$ the solution exists globally for arbitrary large initial value. When $1<p≤p^∗,$ there exists an uniformly bounded global strong solution for small initial value, and the solution decays to zero as $t→ ∞.$ This paper improves and expands the results of [Cong and Liu, Kinet. Relat. Models, 9(4), 2016], in which the parabolic-elliptic case is studied.