1.02 - 1.16 \mu m spectra ( \lambda / \Delta \lambda \sim 7,000 ) of the gravitationally lensed QSO APM 08279+5255 at z _ { em } =3.911 were obtained during the commissioning run of IRCS , the 1 - 5 \micron near-infrared camera and spectrograph for the Subaru 8.2 m Telescope . Strong Mg ii doublet \lambda \lambda 2976,2800 and Fe ii \lambda 2600 , Fe ii \lambda 2587 absorption lines at z _ { abs } =2.974 were clearly detected in the rest-frame UV spectra , confirming the presence of a damped Lyman- \alpha system at the redshift as suggested by Petitjean et al . Also Mg i \lambda 2853 absorption line is probably detected . An analysis of the absorption lines including velocity decomposition was performed . This is a first detailed study of Mg ii absorption system at high redshift ( z > 2.5 ) where the Mg ii doublet shifts into the near-infrared in the observer ’ s frame . The spectra of the lensed QSO pair A and B with 0 \farcs 38 separation were resolved in some exposure frames under excellent seeing condition . We extracted the Mg ii doublet spectra of A and B separately . Although three velocity components ( v \sim -28 , +5 , +45 km s ^ { -1 } ) are known to exist in this Mg ii system ( Petitjean et al . ) , the v \sim +45 km s ^ { -1 } absorption line was not detected toward source B , showing that the +45 km s ^ { -1 } Mg ii cloud lies only in the line of sight to the source A . Our results suggests that the size of the Mg ii absorbing clouds is as small as 200 pc , which corresponds to the separation of A and B at the redshift of the absorber . This is the first direct detection of the small-scale structure of Mg ii clouds at high-redshift , confirming the estimated cloud sizes from photoionization model by Churchill and Charlton .