Low-dimensional Dirac systems offer a versatile platform for quantum-geometric non-linear transport. Hexagonal warping in topological insulator (TI) surface states generates rich momentum-space geometric textures. However, mirror and time-reversal symmetries strictly constrain both the Berry curvature and quantum metric dipoles to zero. Here, we show that proximity-coupling a hexagonally warped TI to a d-wave altermagnet lifts spatial mirror symmetry via the interplay of C_3v warping and d-wave spin-splitting. This symmetry breaking enables finite Berry curvature dipole components, which, however, remain heavily suppressed near charge neutrality, establishing the intrinsic quantum metric dipole as the sole driver of the non-linear Hall response. This scattering-independent response vanishes at μ = 0 and exhibits odd parity under chemical potential inversion. Importantly, rotating the altermagnetic orientation angle φ continuously tunes quantum metric dipole and induces a sign reversal, peaking near φ=0 and vanishing at φ = π/4. Our findings highlight TI-altermagnet interfaces as ideal candidates for gate- and orientation-tunable quantum metric spintronics.