We present our new Atacama Large Millimeter/submillimeter Array (ALMA) observations targeting CO(6-5) emission from three luminous Lyman-break galaxies (LBGs) at z (spec) = 6.0293-6.2037 found in the Subaru/Hyper Suprime-Cam survey, whose [O iii] 88 mu m and [C ii] 158 mu m emissions have been detected with ALMA. We find a marginal detection of the CO(6-5) line from one of our LBGs, J0235-0532, at the similar or equal to 4 sigma significance level and obtain upper limits for the other two LBGs, J1211-0118 and J0217-0208. Our z = 6 luminous LBGs are consistent with the previously found correlation between the CO luminosity and the infrared luminosity. The unique ensemble of the multiple far-infrared emission lines and underlying continuum fed to a photodissociation region model reveals that J0235-0532 has a relatively high density of hydrogen nuclei n (H) that is comparable to those of low-z (U)LIRGs, quasars, and Galactic star-forming regions with high n (H) values, while the other two LBGs have lower n (H) consistent with local star-forming galaxies. By carefully taking account of various uncertainties, we obtain constraints on total gas mass and gas surface density from their CO luminosity measurements. We find that J0235-0532 is located below the Kennicutt-Schmidt (KS) relation, comparable to the z = 5.7 LBG, HZ10, previously detected with CO(2-1). Combined with previous results for dusty starbursts at similar redshifts, the KS relation at z = 5-6 is on average consistent with the local one.