How to utilize topological microcavities to control quantum emission is one of the ongoing research topics in the optical community. In this work, we investigate the emission of quantum emitters in doubly-resonant topological Tamm microcavity, which can simultaneously achieve dual resonances at two arbitrary wavelengths according to the needs of practical application. To achieve the enhancement of quantum emission in such cavities, we have exploited the tunable doubly-resonant modes, in which one of resonant modes corresponds to the pump laser wavelength and the other one is located at the emission wavelength of quantum emitters. Both theoretical and experimental results demonstrate that the pump excitation and emission efficiencies of quantum emitters are greatly enhanced. The main physical mechanism can be explained by the doubly-resonant cavity temporal coupled-mode theory. Furthermore, we observe the faster emission rate and the higher efficiency of unidirectional quantum emission, which have promising applications in optical detection, sensing, filtering, and light-emitting devices.