Close supermassive binary black holes (SMBBHs) with separations less than about 0.1 parsec are expected to be Nano-Hertz gravitational wave sources. SMBBH systems should exhibit periodic variability. However, periodic variability in radio-loud quasars may be interpreted with the jet model. Here we report the detection of a robust periodic signal in the optical variability of the radio-quiet quasar PG 0923+201 with an observed period of 726.8±4.7 days, obtained from the sinusoid-like light curve of a temporal baseline of about 9 years. This periodicity is probably from a close SMBBH with a total mass of 109.3 solar masses and a separation of about 0.01 parsec, implying relativistic orbital speeds. Such a system has passed through the well-known “final parsec problem” of SMBBH systems, and the Nano-Hertz gravitational wave radiation becomes significant. The ratio of the separation between these two black holes to the broad-line region size is about 0.1. A close SMBBH is also suggested by this small ratio and the spectral properties of Balmer broad lines in this quasar. This radio-quiet quasar is a candidate emitter of Nano-Hertz gravitational waves at a frequency of about 30 Nano-Hertz.