We present deep (265 ks) Chandra X-ray observations of PSO J352.4034-15.3373, a quasar at z = 5.831 that, with a radio-to-optical flux ratio of R > 1000, is one of the radio-loudest quasars in the early universe and is the only quasar with observed extended radio jets of kiloparsec scale at z greater than or similar to 6. Modeling the X-ray spectrum of the quasar with a power law, we find a best fit of Gamma = 1.99(-0.28)(+0.29), leading to an X-ray luminosity of L2-10 = 1.26(-0.33)(+0.45) x 10(45) erg s(-1) and an X-ray to UV brightness ratio of alpha(OX) = -1.45 +/- -0.11. We identify a diffuse structure 50 kpc (similar to 8 '') to the NW of the quasar along the jet axis that corresponds to a 3 sigma enhancement in the angular density of emission and can be ruled out as a background fluctuation with a probability of P = 0.9985. While with few detected photons the spectral fit of the structure is uncertain, we find that it has a luminosity of L2-10 similar to 10(44) erg s(-1). These observations therefore potentially represent the most distant quasar jet yet seen in X-rays. We find no evidence for excess X-ray emission where the previously reported radio jets are seen (which have an overall linear extent of 0.'' 28), and a bright X-ray point source located along the jet axis to the SE is revealed by optical and NIR imaging to not be associated with the quasar.