In this paper, a learning-based approach is proposed for optimizing downlink beamforming in multiple-input multiple-output (MIMO) systems that employ continuous aperture arrays (CAPAs) at both the base station (BS) and the user. Beamforming in such systems is a spatially continuous function that maps a coordinate on the CAPA to a corresponding beamforming weight. We first propose an implicit neural representation (INR), termed BeaINR, to parameterize this function directly. Further, noting that the optimal beamforming function can be expressed as a weighted integral of the channel response function, we propose a second INR, CoefINR, to represent the weighting coefficient function, which indirectly optimizes the beamforming function. Simulation results show that the proposed INR-based methods achieve comparable or higher spectral efficiency (SE) than the considered baselines, while requiring substantially lower inference latency. Moreover, CoefINR reduces training complexity and improves frequency generalizability relative to BeaINR by leveraging the optimal beamforming structure.