Josephson junctions are typically characterized by a single phase difference across two superconductors. This conventional two-terminal Josephson junction can be generalized to a multi-terminal device where the Josephson energy contains terms with contributions from multiple independent phase variables. Such multi-terminal Josephson junctions (MTJJs) are being considered as platforms for engineering effective Hamiltonians with non-trivial topologies, such as Weyl crossings and higher-order Chern numbers. This approach offers unique possibilities that are complementary to phenomena attainable in bulk crystals, including topological states in more than three dimensions and real-time gate-tunability of the Hamiltonians. However, these prospects rely on the ability to create MTJJs with non-classical multi-terminal couplings in which only a handful of quantum modes are populated. Here, we demonstrate these requirements by using a three-terminal Josephson junction fabricated on selective-area-grown (SAG) PbTe nanowires. We observe signatures of a $\pi$-shifted Josephson effect, consistent with inter-terminal couplings mediated by four-particle quantum states called Cooper quartets. We further observe supercurrent co-existent with a non-monotonic evolution of the conductance with gate voltage, indicating transport mediated by a few quantum modes in both two- and three-terminal devices. These results establish a platform for investigations of topological Hamiltonians based on Andreev bound states.
