We report the highest-redshift detection of [O i] 63 $\mu$m from a luminous quasar, J2054$-$0005, at $z=6.04$ based on the Atacama Large Millimeter/sub-millimeter Array (ALMA) Band 9 observations. The [O i] 63 $\mu$m line luminosity is $(4.5\pm 1.5) \times 10^{9} L_{\odot }$, corresponding to the [O i] 63 $\mu$m-to-far-infrared luminosity ratio of $\approx$6.7 $\times 10^{-4}$, which is consistent with the value obtained in the local Universe. Remarkably, [O i] 63 $\mu$m is as bright as [C ii] 158 $\mu$m, resulting in the [O i]-to-[C ii] line luminosity ratio of $1.3\pm 0.5$. Based on a careful comparison of the luminosity ratios of [O i] 63 $\mu$m, [C ii] 158 $\mu$m, and dust continuum emission to models of photodissociation regions, we find that J2054$-$0005 has a gas density $\log (n_{\rm H}/{\rm cm}^{-3}) = 3.7\pm 0.3$ and an incident far-ultraviolet radiation field of $\log (G/G_{\rm 0}) = 3.0\pm 0.1$, showing that [O i] 63 $\mu$m serves as an important coolant of the dense and warm gas in J2054$-$0005. A close examination of the [O i] and [C ii] line profiles suggests that the [O i] line may be partially self-absorbed; however, deeper observations are needed to verify this conclusion. Regardless, the gas density and incident radiation field are in broad agreement with the values obtained in nearby star-forming galaxies and objects with [O i] 63 $\mu$m observations at $z=1$–3 with the Herschel Space Observatory. These results demonstrate the power of ALMA high-frequency observations targeting [O i] 63 $\mu$m to examine the properties of photodissociation regions in high-redshift galaxies.