One leading question for the application of graphene in nanoelectronics is how electronic properties depend on the size at the nanoscale. Direct observation of the quantized electronic states is central to conveying the relationship between electronic structures and local geometry. Scanning tunneling spectroscopy was used to measure differential conductance dI/dV patterns of nanometer-size graphene islands on an Ir(111) surface. Energy-resolved dI/dV maps clearly show a spatial modulation, indicating a modulated local density of states due to quantum confinement, which is unaffected by the edge configuration. We establish the energy dispersion relation with the quantized electron wave vector obtained from a Fourier analysis of dI/dV maps. The nanoislands preserve the Dirac Fermion properties with a reduced Fermi velocity.