The capacitance of a nanojunction formed by a scanning tunneling microscope (STM) tip and a two-dimensional gold cluster was measured through the single electron tunneling spectroscopy of a double-barrier tunnel junction. By decreasing the STM tip-cluster separation, it was observed that the capacitance first increases and then decreases at short separation. This characteristic clearly deviates from the classical behavior and provides evidence for potential quantum effects on the capacitance.