<p>Using the integral transformation method, this study investigates the dynamic propagation problem of a moving crack in a functionally graded one-dimensional hexagonal quasicrystal strip. A Yoffe-type constant-velocity crack model was adopted. The boundary value problem of the partial differential equation was transformed into two pairs of dual integral equations through Fourier cosine transform, which are then solved by the Copson method. The stress field distribution near the crack tip was obtained. Expressions for the dynamic stress intensity factors (DSIFs) and the dynamic energy release rate of the phonon and phason fields were further derived. A numerical simulation was conducted to analyze the influences of material gradient parameters, crack velocity, crack length, strip thickness, phonon–phason coupling coefficient, and external load on the fracture behavior. The research results are expected to provide a theoretical basis for the fracture performance evaluation and gradient optimization design of functionally graded quasicrystals under dynamic loading.</p>