Many quiescent galaxies discovered in the early Universe by JWST raise fundamental questions on when and how these galaxies became and stayed quenched. Making use of the latest version of the semianalytic model GAEA that provides good agreement with the observed quenched fractions up to z similar to 3, we make predictions for the expected fractions of quiescent galaxies up to z similar to 7 and analyze the main quenching mechanism. We find that in a simulated box of 685 Mpc on a side, the first quenched massive (M star similar to 1011 M circle dot), Milky Way-mass, and low-mass (M star similar to 109.5 M circle dot) galaxies appear at z similar to 4.5, z similar to 6.2, and before z = 7, respectively. Most quenched galaxies identified at early redshifts remain quenched for more than 1 Gyr. Independently of galaxy stellar mass, the dominant quenching mechanism at high redshift is accretion disk feedback (quasar winds) from a central massive black hole, which is triggered by mergers in massive and Milky Way-mass galaxies and by disk instabilities in low-mass galaxies. Environmental stripping becomes increasingly more important at lower redshift.