Thermal transport of quantum magnets has elucidated the nature of low energy elementary excitations and complex interplay between those excited states via strong scattering of thermal carriers. BiCu2PO6 is a unique frustrated spin-ladder compound exhibiting highly anisotropic spin excitations that contain both itinerant and localized dispersion characters along the b- and a-axes respectively. Here, we investigate thermal conductivity. of BiCu2PO6 under high magnetic fields (H) of up to 30 tesla. A dip-feature in., located at similar to 15 K at zero-H along all crystallographic directions, moves gradually toward lower temperature (T) with increasing H, thus resulting in giant suppression by a factor of similar to 30 near the critical magnetic field of H-c congruent to 23.5 tesla. The giant H- and T-dependent suppression of kappa can be explained by the combined result of resonant scattering of phononic heat carriers with magnetic energy levels and increased phonon scattering due to enhanced spin fluctuation at Hc, unequivocally revealing the existence of strong spin-phonon coupling. Moreover, we find an experimental indication that the remaining magnetic heat transport along the b-axis becomes almost gapless at the magnetic quantum critical point realized at H-c.