We study Family–Vicsek scaling of charge, spin, and energy fluctuations in the one-dimensional Hubbard model at infinite temperature. Using a quantum generating function approach, we compute time-dependent cumulants of transferred conserved quantities and analyze how the corresponding roughness depends on subsystem size and time. We start by focusing on a single interface at half the chain and determine the transport exponents. Then we turn to fluctuations of a small finite interval and study the Family–Vicsek universality of fluctuations over an extended timescale. We find that the long-time scaling behavior is controlled by integrability. In the free limit, charge, spin, and energy all display ballistic transport. In the interacting integrable Hubbard chain, charge and spin cross over to a KPZ-type scaling regime at the level of the second cumulant, while the energy sector remains ballistic. For the representative integrability-breaking perturbation studied here, the accessible long-time dynamics becomes diffusive in all sectors. In every case we also observe a short-time microscopic regime with apparently universal ballistic growth before the hydrodynamic scaling window sets in. The Family–Vicsek setup allows us to determine the growth, the saturation as well as the dynamical exponents.