Point defect-based single-photon emitters (SPEs) in GaN have aroused a great deal of interest due to their room-temperature operation, narrow line width and high emission rate. The room-temperature SPEs at the telecommunication bands have also been realized recently by localized defects in GaN in experiments, which are highly desired for the practical applications of SPEs in quantum communication with fiber compatibility. However, the origin and underlying mechanism of the SPEs remain unclear to date. Herein, our first-principle calculations predict and identify an intrinsic point defect NGa in GaN that owns a zero-phonon line (ZPL) at telecommunication windows. By tuning the triaxial compressive strain of the crystal structure, the ZPL of NGa can be modulated from 0.849 eV to 0.984 eV, covering the fiber telecommunication windows from the O band to the E band. Besides the ZPL, the formation energy, band structure, transition process and lifetime of the SPEs under different strains are investigated systematically. Our work gives insight into the emission mechanism of the defect SPEs in GaN and also provides effective guidance for achieving wavelength-tunable SPEs working in fiber telecommunication windows.