We discovered H alpha absorption in the broad H alpha emission line of an unusual broad absorption line quasar, SDSS J083942.11+380526.3, at z 2: 318, through near-infrared spectroscopy with the Cooled Infrared Spectrograph and Camera for OHS (CISCO) on the Subaru telescope. The presence of nonstellar H alpha absorption is known only in the Seyfert galaxy NGC 4151 to date; thus, our discovery is the first case for quasars. The H alpha absorption line is blueshifted by 520 km s(-1) relative to the H alpha emission line, and its redshift almost coincides with those of UV low-ionization metal absorption lines. The width of the H alpha absorption (similar to 340 km s(-1)) is similar to those of the UV low-ionization absorption lines. These facts suggest that the H alpha and low-ionization metal absorption lines are produced by the same low-ionization gas, which has a substantial amount of neutral gas. The column density of the neutral hydrogen is estimated to be similar to 10(18) cm(-2) by assuming a gas temperature of 10,000 K from the analysis of the curve of growth. The continuum spectrum is reproduced by a reddened [E(B-V) similar to 0.15mag for the SMC-like reddening law] composite quasar spectrum. Furthermore, the UV spectrum of SDSS J083942.11+ 380526.3 shows a remarkable similarity to that of NGC 4151 in its low state, suggesting that the physical condition of the absorber in SDSS J083942.11+ 380526.3 is similar to that of NGC 4151 in the low state. As proposed for NGC 4151, SDSS J083942.11+ 380526.3 may also be seen through the edge of the obscuring torus.