This paper proposes an efficient analytical modeling method for the air-gap flux density in hydro-generators under eccentricity faults. The method specifically addresses the multi-factor coupling characteristics induced by the combined effects of stator slotting, salient-pole rotor structure, and eccentricity. Based on geometric analytical derivation, individual models for air-gap length considering stator slotting, rotor salient-pole structure, and eccentricity are established, and an analytical expression for the air-gap length under coupled conditions is derived. By incorporating the spatial distribution characteristics of the magnetomotive force, a mathematical model for no-load air-gap flux density is constructed, enabling the rapid calculation of air-gap magnetic field distribution. Finite element verification conducted on an 84-slot, 10-pole hydro-generator demonstrates that the proposed method accurately reflects the periodic fluctuations and amplitude variations in the magnetic flux density under both normal and eccentric conditions. Compared with the finite element method (FEM), the analytical approach significantly enhances computational efficiency while maintaining high accuracy, providing effective theoretical support for the structural optimization and eccentricity fault diagnosis of hydro-generators.