A combination of graphene-like electrodes and ionic liquid (IL) electrolytes has emerged as a viable and attractive choice for electrochemical double layer (EDL) capacitors. Based on combined classical molecular dynamics and density functional theory calculations, we present the interfacial capacitance between planar graphene and [BMIM][PF6] IL, with particular attention to the relative contributions of the electric double layer capacitance at the graphene/IL interface and the quantum capacitance of graphene. The microstructure of [BMIM][PF6] near the graphene electrode with varying charge densities are investigated to provide a molecular description of EDLs, including BMIM/PF6 packing and orientation, cation-anion segregation, and electrode charge screening. Although the IL interfacial structures exhibit an alternative cation/anion layering extending a few nanometers, the calculated potential profiles provide evidence of one-ion thick compact EDL formation. The capacitance-potential curve of the EDL is convex- or bell-shaped, whereas the quantum capacitance of graphene is found to have concave- or U-shaped characteristics with a minimum of nearly zero. Consequently, we find that the total interfacial capacitance exhibits a U-shaped trend, consistent with existing experimental observations at a typical carbon/IL interface. Our work highlights the importance of the quantum capacitance in the overall performance of graphene-based EDL capacitors.