We describe the sensitivity of the IceCube Neutrino Observatory to the creation of axions in a Type II core-collapse supernova (SN II) in the Galaxy. Axions are a light dark matter candidate, and their existence is well-motivated as a solution to the strong CP problem. In a supernova, axions behave much like neutrinos, efficiently removing energy from the explosion and cooling the system. As a result, neutrino production in an SN II is suppressed in the presence of axions, mainly affecting the seconds-long tail of the SN II neutrino light curve. The IceCube Observatory is sensitive to large numbers of MeV neutrinos from an SN II, which produce a collective rise in the hit rates of all photomultipliers in the detector. We present a shape analysis that can be used with IceCube data to discriminate the axion production scenario from standard SN II models.