We report periodic oscillations in the 15-year-long optical light curve of the gravitationally lensed quasar Q J0158−4325 at z_s = 1.29. The signal is enhanced during a high magnification microlensing event of the quasar that the fainter lensed image, B, underwent between 2003 and 2010. We measure a period of P_o = 172.6 ± 0.9 days, which translates to 75.4 ± 0.4 days in the quasar frame. The oscillations have a maximum amplitude of 0.26 ± 0.02 mag and decrease concurrently with the smooth microlensing amplitude. We explore four scenarios to explain the origin of the periodicity: (1) the high magnification microlensing event is due to a binary star in the lensing galaxy, (2) Q J0158−4325 contains a supermassive binary black hole system in its final dynamical stage before merging, (3) the quasar accretion disk contains a bright inhomogeneity in Keplerian motion around the black hole, and (4) the accretion disk is in precession. Of these four scenarios, only a supermassive binary black hole can account for both the short observed period and the amplitude of the signal, through the oscillation of the accretion disk towards and away from high-magnification regions of a microlensing caustic. The short measured period implies that the semi-major axis of the orbit is ∼10⁻³ pc and that and the coalescence timescale is t_coal ∼ 1000 yr, assuming that the decay of the orbit is solely powered by the emission of gravitational waves. The probability of observing a system so close to coalescence, in a sample of only 30 monitored lensed quasars, suggests either a much larger population of supermassive binary black holes than predicted or, more likely, that some other mechanism significantly increases the coalescence timescale. Three tests of the binary black hole hypothesis include: (i) the recurrence of oscillations in photometric monitoring during any future microlensing events in either image, (ii) spectroscopic detection of Doppler shifts (up to ∼0.01c) associated with optical emission in the vicinity of the black holes, and (iii) the detection of gravitational waves through pulsar timing array experiments, such as the Square Kilometre Array, which will have the sensitivity to detect the ∼100 nano-hertz emission.