A dyadic Green's function of a wedge with anisotropic impedance faces, excited by an electric dipole source, is derived for the paraxial region where the source and observation points are in proximity to the apex but widely separated. The principal anisotropy axis is the edge axis, and surface impedances parallel and transverse to this axis are considered. Following a separation of variables derivation, final dyadics involve eigenfunction solutions over an angular wave number and a longitudinal spectral integral, which is evaluated asymptotically assuming that k|z - z| is large. It is observed that derived forms reveal three distinct scattering mechanisms: edge-guided waves, surface waves, and guided waves in the classical sense. Numerical simulations limited to paraxial region show that edge-guided and guided-wave terms are dominant at points away from the wedge surface, whereas surface waves are dominant near impedance surfaces. Both capacitive, inductive, and mixed (one face capacitive and the other inductive) reactive surface impedances are numerically analyzed. The resulting expressions can be used in the analysis of antennas located near the apex of a wedge and electromagnetic scattering from artificially hard and soft surfaces.