Altermagnetism is a recently established class of magnetic order that combines fully compensated moments with momentum-dependent spin splitting, yet identifying its spectroscopic fingerprints remains an open challenge. In this work, we investigate finite-momentum two-magnon excitations in a two-dimensional d-wave altermagnet by calculating the two-magnon light scattering intensity in different polarization channels. Using linear spin-wave theory and further incorporating the leading 1/S quantum corrections, we demonstrate that the characteristic magnon splitting of altermagnets shifts the upper edge of the two-magnon continuum and the corresponding spectral features to higher energy at X=(π,0) relative to the conventional antiferromagnet, by an amount linear in the exchange anisotropy |δ J₂|, while leaving the response at the Brillouin-zone center unchanged. Although magnon-magnon interactions strongly redistribute spectral weight toward lower energies, the energy scale associated with the high-energy momentum-selective reconstruction remains robust. The interactions additionally generate a pronounced low-energy two-peak structure that has no counterpart within linear spin-wave theory. At K=(π/2,π/2), we show that the interacting two-magnon resonance is twofold degenerate in the conventional antiferromagnetic phase, while a finite altermagnetic exchange anisotropy lifts this degeneracy, producing two peaks whose separation is linear in |δ J₂|. The characteristic energy scales underlying both features are intrinsic to the two-magnon sector rather than to a specific scattering operator. These findings highlight the potential of finite-momentum two-magnon spectroscopy for identifying altermagnetic order in insulating magnets and motivate momentum-resolved resonant inelastic x-ray scattering studies of candidate altermagnetic materials.