We explore the Matsubara quasiparticle fraction and the pseudogap of the two-dimensional Hubbard model with the dynamical cluster quantum Monte Carlo method. The character of the quasiparticle fraction changes from non-Fermi-liquid, to marginal Fermi liquid, to Fermi liquid as a function of doping, indicating the presence of a quantum critical point separating non-Fermi-liquid from Fermi-liquid character. Marginal Fermi-liquid character is found at low temperatures at a very narrow range of doping where the single-particle density of states is also symmetric. At higher doping the character of the quasiparticle fraction is seen to cross over from Fermi liquid to marginal Fermi liquid as the temperature increases.