In this work, the elastic 4 × 4 Green tensor of one-dimensional quasicrystals is given and has phonon, phason and phonon-phason coupling components. Using the residue method, a closed-form expression of the elastic 4 × 4 Green tensor for one-dimensional hexagonal quasicrystals of Laue class 10, which possess 10 independent material constants, is derived. The 10 independent components of the obtained 4 × 4 Green tensor are numerically presented in contour plots revealing features of anisotropy as well as the interesting result that the phason component of the Green tensor has the strongest contribution in comparison with all the other components. In the case of vanishing phonon-phason coupling, the phonon part of the derived Green tensor reproduces Kröner’s well-known elastic 3 × 3 Green tensor for hexagonal crystals. The analytical closed-form expression of the derived Green tensor provides an advantage for efficient computational calculations in various applications.