We present deep Keck/MOSFIRE near-infrared spectroscopy of a strong Ly alpha emitting source at z = 6.1292, HSC J142331.71-001809.1, which was discovered by the SHELLQS program from imaging data of the Subaru Hyper Suprime-Cam (HSC) survey. This source is one of five objects that show narrow (FWHM < 500 km s(-1)) and prominent (Ly alpha > 1044 erg s(-1)) Ly alpha emission lines at absolute 1450 angstrom continuum magnitudes of M-1450 similar to-22 mag. Its rest-frame Ly alpha equivalent width (EW) is 370 +/- 30 angstrom. In the 2 hr Keck/MOSFIRE spectrum in Y band, the high-ionization C IV lambda lambda 1548,1550 doublet emission line was clearly detected with FWHM = 120(-20)(+20) km s(-1) and a total rest-frame EW of 37(-5)(+6) angstrom. We also report the detection of weak continuum emission, and the tentative detection of O III]lambda lambda 1661,1666 in the 4 hr J-band spectrum. Judging from the UV magnitude, line widths, luminosities, and EWs of Ly alpha and C IV, we suggest that this source is a reionization-era analog of classical type-II AGNs, although there is a possibility that it represents a new population of AGN/galaxy composite objects in the early universe. We compare the properties of J1423-0018 to intermediate-redshift type-II AGNs and C IV emitters seen in z = 6-7 galaxy samples. Further observations of other metal emission lines in the rest-frame UV or optical, along with X-ray follow-up observations of the z = 6-7 narrow-line quasars, are needed for more robust diagnostics and to determine their nature.