Controlled quantum mechanical motion of trapped atomic ions can be used to simulate and explore collective quantum phenomena and to process quantum information. Groups of cold atomic ions confined in an externally applied trapping potential self-organize into “Coulomb crystals” due to their mutual electrostatic repulsion. The motion of the ions in these crystals is strongly coupled, and most eigenmodes of motion involve multiple ions. While this enables studies of many-body physics, it limits the flexibility and tunability of the system as a quantum platform. Here, we demonstrate an array of trapped ions in individual trapping sites whose motional modes can be controllably coupled and decoupled by locally tuning the confining potential for each ion. A single motional quantum, or phonon, can be coherently shared among two or three ions located at the vertices of an equilateral triangle 30 µm on a side. We tune the motional eigenmodes about closed contours in configuration space and observe that if the contour adiabatically encircles a conical intersection present in the motional eigenvalue surfaces, the single-phonon wavefunction acquires a topological Berry phase that is independent of the contour shape. We characterize these phases by single-phonon interference and study how phase changes occur when the motional mode tuning becomes non-adiabatic. Our results show that precise, individual quantum control of ion motion in a two-dimensional array provides unique access to quantum multi-body effects.