Scanning tunneling spectroscopy (STS) was used to measure local differential conductance (dI/dV) spectra on nanometer-size graphene islands on an Ir(111) surface. Energy resolved dI/dV maps clearly show a spatial modulation, which we ascribe to a modulated local density of states due to quantum confinement. STS near graphene edges indicates a position dependence of the dI/dV signals, which suggests a reduced density of states near the edges of graphene islands on Ir(111).