It is of scientific importance to obtain graphene quantum dots (GQDs) with narrow-size distribution in order to unveil their size-dependent structural and optical properties, thereby further to explore the energy band diagram of GQDs. Here, a soft-template microwave-assisted hydrothermal method to prepare GQDs with diameters less than 5 nm ± 0.55 nm is reported. The size-dependent photoluminescence (PL) quantum yield (QY) decay lifetime and electron energy loss spectroscopy (EELS) of the GQDs are studied systematically. The QY of the GQDs with an average diameter of 2 nm is the highest (15%) among all the samples investigated and the QY decreases with increasing diameter of the GQDs. The size-dependence of the PL decay lifetime is also observed. The result suggests that spatial confinement effects related to radiative relaxation play an important role in the size-dependent decay lifetime. A realistic energy band diagram of the GQDs is deduced from the experimental results.