Quasars at early redshifts (z > 6) with companion galaxies offer unique insights into the growth and evolution of the first supermassive black holes. Here, we report on a 150 ks Chandra observation of PSO.J308.0416-21.2339, a z = 6.23 quasar with a merging companion galaxy identified in [C II] and rest-frame UV emission. With 72.3(-8.6)(+9.6) net counts, we find that PSO.J308.0416-21.2339 is powerful (L-X = 2.31(-0.76)(+1.14) x 10(45) erg s(-1) cm(-2) in rest-frame 2.0-10.0 keV) yet soft (spectral power-law index Gamma = 2.39(-0.36)(+0.37) and optical-to-X-ray slope alpha(OX) = -1.41 +/- 0.11). In addition, we detect three hard-energy photons 2 ''.0 to the west of the main quasar, cospatial with the brightest UV emission of the merging companion. As no soft-energy photons are detected in the same area, this is potentially indicative of a highly obscured source. With conservative assumptions, and accounting for both background fluctuations and the extended wings of the quasar's emission, these photons only have a probability P = 0.021 of happening by chance. If confirmed by deeper observations, this system is the first high-redshift quasar and companion individually detected in X-rays and is likely a dual active galactic nucleus.