We present the results from a ~ 500 ks Chandra observation of the z = 6.31 QSO SDSS J1030 + 0524. This is the deepest X-ray observation to date of a z ~ 6 QSO. The QSO is detected with a total of 125 net counts in the full (0.500A0–7 keV) band and its spectrum can be modeled by a single power-law model with photon index of Γ = 1.81 ± 0.18 and full band flux of f = 3.95 × 10⁻¹⁵ erg s⁻¹ cm⁻². When compared with the data obtained by XMM-Newton in 2003, our Chandra observation in 2017 shows a harder (ΔΓ ≈ −0.6) spectrum and a 2.5 times fainter flux. Such a variation, in a timespan of ~ 2 yr rest-frame, is unexpected for such a luminous QSO powered by a > 10⁹M_⨀ black hole. The observed source hardening and weakening could be related to an intrinsic variation in the accretion rate. However, the limited photon statistics does not allow us to discriminate between an intrinsic luminosity and spectral change, and an absorption event produced by an intervening gas cloud along the line of sight. We also report the discovery of diffuse X-ray emission that extends for 30″ × 20″ southward of the QSO with a signal-to-noise ratio (S/N) of approximately six, hardness ratio of HR = 0.03^+0.20_−0.25, and soft band flux of f_{0.5– keV} = 1.1^+0.3_−0.3 × 10⁻¹⁵ erg s⁻¹ cm⁻² , that is not associated to a group or cluster of galaxies. We discuss two possible explanations for the extended emission, which may be either associated with the radio lobe of a nearby, foreground radio galaxy (at z ≈ 1 – 2), or ascribed to the feedback from the QSO itself acting on its surrounding environment, as proposed by simulations of early black hole formation.