Magneto-active elastomers exhibit large, strongly coupled deformations under combined mechanical loading and applied magnetic fields, with effective behavior governed by heterogeneous particle microstructures and the near-incompressible response of the polymer matrix. This work presents a fully coupled computational homogenization framework in which the microscale mechanical and magnetostatic boundary-value problems are solved monolithically on a periodic representative volume element. The approach uses a mixed finite-element setting with Lagrangian displacements and a magnetic vector potential to obtain a divergence-free induction field and to enforce periodic constraints for both the mechanical and magnetic unknowns. Near-incompressibility is treated via an element-wise J-bar stabilization. The volumetric response is controlled by the cell-averaged dilatation while the isochoric response is evaluated from a scaled deformation measure, improving robustness at finite strain without introducing an additional pressure field. The constitutive behavior follows from an additive Helmholtz free-energy decomposition combining a Yeoh-type hyperelastic contribution, a vacuum magnetic energy term, and a saturation-type magnetization potential activated in the particle phase. Consistent linearization is achieved through automatic differentiation of the free energy, yielding all required tangent operators including magneto-mechanical coupling terms. Benchmark and demonstration studies, including a single-inclusion comparison between hexahedral and tetrahedral discretizations, particle-chain microstructures, and random particle ensembles, highlight stable simulation performance, microstructuredependent coupling effects, and reliable extraction of homogenized stress and magnetic response quantities for multiscale analyses. Future work will integrate physics augmented machine learning surrogates to accelerate the microscale simulations, enabling efficient inverse design and optimization of magnetoactive composites.