The harmonic oscillator is one of the simplest physical systems but also one of the most fundamental. It is ubiquitous in nature, often serving as an approximation for a more complicated system or as a building block in larger models. Realizations of harmonic oscillators in the quantum regime include electromagnetic fields in a cavity(1-3) and the mechanical modes of a trapped atom(4) or macroscopic solid(5). Quantized interaction between two motional modes of an individual trapped ion has been achieved by coupling through optical fields(6), and entangled motion of two ions in separate locations has been accomplished indirectly through their internal states(7). However, direct controllable coupling between quantized mechanical oscillators held in separate locations has not been realized previously. Here we implement such coupling through the mutual Coulomb interaction of two ions held in trapping potentials separated by 40 mm (similar work is reported in a related paper(8)). By tuning the confining wells into resonance, energy is exchanged between the ions at the quantum level, establishing that direct coherent motional coupling is possible for separately trapped ions. The system demonstrates a building block for quantum information processing and quantum simulation. More broadly, this work is a natural precursor to experiments in hybrid quantum systems, such as coupling a trapped ion to a quantized macroscopic mechanical or electrical oscillator(9-13).