Quasilinear theory of a weakly turbulent quantum Fermi liquid is presented. Landau's linear theory of Fermi liquids is generalized by consideration of weak nonlinear regime. A newly derived kinetic equation of the Fermi particles is used to derive a slowly varying distribution function f0, which satisfies the diffusion equation. It is shown that the magnitude of the diffusion coefficient D depends on weak interactions between atoms and the de Broglie waves diffraction.