The predominantly d_(x²-y²)-wave pairing symmetry of cuprate superconductors is well established, yet its microscopic manifestation in local Cooper-pair tunneling remains less explored. Here, we investigate Bi_(1.2)Pb_(0.8)Sr₂CaCu₂O_(8+δ), using Josephson scanning tunneling spectroscopy with a conventional s-wave superconducting Pb tip. We observe a zero-bias conductance enhancement whose approximately quadratic dependence on junction conductance is consistent with phase-diffusive Josephson tunneling. Atomic-resolution conductance maps and site-resolved spectra reveal a pronounced intra-unit-cell modulation, with enhanced Josephson response near oxygen sites and strong suppression near copper sites and along diagonal directions. Microscopic calculations incorporating the bilayer electronic structure and realistic Wannier-based tunneling matrix elements reproduce the observed spatial pattern through interference between pairing contributions of opposite signs. Our results reveal how the sign-changing d-wave superconducting order parameter manifests itself in atomic-scale Josephson tunneling, providing complementary evidence for unconventional pairing symmetry in cuprates.