Altermagnets are rapidly emerging as a highly promising platform for spintronics, yet dynamically controlling their spin responses remains a fundamental challenge. In this work, we demonstrate that introducing periodic optical driving and non-Hermiticity provides a powerful route to achieve tunable control over these systems. We derive a general analytical expression for nonlinear spin currents in non-Hermitian phases with a spectral line gap, revealing that the intrinsic response cleanly separates into quantum metric, Berry curvature, and Berry connection dipole contributions. Applying this formalism to a Floquet non-Hermitian d-wave altermagnet, we uncover that the nonlinear spin conductivity is overwhelmingly dominated by the bare quantum metric. Furthermore, we show that the optical field's polarization can actively tune – and even strictly reverse – the direction of both longitudinal and transverse spin currents. Our work establishes a quantum geometric framework for the optical manipulation of nonlinear spin transport in advanced magnetic materials.