We systematically investigate the microscopic origins of the intrinsic anomalous Hall effect (AHE) and spin Hall effect (SHE) in the rare-earth transition-metal (RE–M) ferrimagnet {GdCo}5 using first-principles DFT+U+J calculations combined with Berry-curvature analysis. The Hall-active states near the Fermi energy in the ground-state antiparallel ferrimagnetic configuration are predominantly Co-derived. However, changes in the relative Gd–Co magnetic alignment substantially reconstruct these electronic states, modifying the relevant band dispersions and interband energy separations and thereby redistributing the Berry curvature and Hall response. Based on this mechanism, we elucidate the microscopic origin of the pronounced suppression of the anomalous Hall conductivity (AHC) during the field-induced spin-flop transition, showing that the dominant reduction occurs in the first stage of spin reorientation, despite the Gd–Co antiparallel alignment remaining largely preserved. At the Fermi energy, the AHC reaches its maximum magnitude of 885 {( cm)}^{-1} in the antiparallel ground state. Furthermore, we show that {GdCo}5 exhibits a large intrinsic spin Hall conductivity (SHC), with the dominant tensor components reaching {zx}^{y = 1558 (/e){( cm)}}{-1} and {yx}^{z = 731 (/e){( cm)}}{-1} at the Fermi level, yielding a substantial spin Hall angle up to 0.13. In addition, selected SHC tensor components can be strongly modulated and even reversed in sign through magnetization rotation and changes in the relative Gd–Co alignment, reflecting the sensitivity of the spin Hall response to the magnetic Laue group symmetry and relative sublattice configuration. Our findings reveal a microscopic connection between magnetic configuration, electronic-structure reconstruction, and Berry-curvature-driven transport in {GdCo}_5, and highlight its potential for exploring magnetization-dependent intrinsic spin-current generation.