The pseudo-Stroh formulism can transform the governing equations of multi-field coupling materials such as quasicrystals into a linear eigensystem, enabling the exact solution of multilayered structures with simply-supported boundary conditions. This provides an important reference for various numerical and experimental methods of quasicrystal beams in engineering practice. In this paper, the free vibration and bending problems of one-dimensional (1D) hexagonal quasicrystal (QC) laminated beams with functional gradients are investigated using the pseudo-Stroh formula. A simply-supported QC laminated beam is modeled, and the transfer matrix method is used to derive the exact solutions for natural frequency of free vibration and bending deformation displacements of the beam under simply-supported boundary conditions. The obtained results are compared with the existing ones to verify the accuracy and precision of the presented model. Numerical examples are provided to show the effects of high span ratio, layer thickness ratio, and functional gradient coefficient on the natural frequency, bending deformation, and mode shape of simply-supported 1D QC laminated beams under two different stacking sequences. The results show that natural frequency increases with the increase of functional gradient coefficient. Phonon displacement decreases while phason displacement increases with the increase of functional gradient coefficient under the two stacking sequences. Functional gradient coefficient and stacking sequence minimally affect phonon displacement modes but significantly impact phason displacement modes. Moreover, they notably affect phason stresses compared to phonon stresses in QC laminated beams. Thus, the optimal natural frequency and deformation displacement of a QC beam can be achieved by adjusting geometric size, stacking sequence, and functional gradient coefficient of the layered beam. These findings can provide theoretical references for various numerical methods and experimental studies on QC beams.