Nuclear quantum effects (NQEs) are highly important for understanding a host of kinetic processes that occur with the participation of H (e.g., H adsorption, diffusion, permeation, and trapping in materials). In this paper, ab initio path integral molecular dynamics simulations were used to investigate NQEs on the lattice diffusion of H in common face-centered cubic (fcc) metals such as Al, Ag, and Cu over a wide temperature range of 75-1200 K (75-900 K for Al). We determined that the dependence of H diffusivities on temperature in Ag and Cu has a "reversed-S" shape on Arrhenius plots, as confirmed for fcc Pd in our recent study [H. Kimizuka et al., Phys. Rev. B 100, 024104 (2019)]. This result illustrates that the phenomenon is common in many fcc metals in which H atoms prefer to occupy octahedral sites. On the other hand, in the case of Al, in which H atoms prefer to occupy tetrahedral sites, the dependence of H diffusivities on temperature exhibits a familiar "C" shape. Such counterintuitive behavior is ascribed to differences in the dependence on temperature of the activation barriers for H migration between both types of fcc metals; this is due to the NQEs involving a competition between deceleration of H migration, which becomes effective at high temperatures because of zero-point vibrations, and acceleration of H migration, which becomes effective at low temperatures because of quantum tunneling. The dominance of the two mechanisms is determined by the coupling of the NQEs and the site preference of H depending on the metal. This finding has important implications for the interpretation of kinetic processes involving the crossover from classical to quantum behavior of H atoms jumping between different types of interstitial sites.