We present the results of deep spectroscopy for the central region of the cluster lens SDSS J1004+4112 with the Subaru telescope. A secure detection of an emission line of the faint blue stellar object (component E) near the center of the brightest c
luster galaxy (G1) confirms that it is the central fifth image of the lensed quasar system. In addition, we measure the stellar velocity dispersion of G1 to be sigma_* = 352+-13 km/s. We combine these results to obtain constraints on the mass M_BH of the putative black hole (BH) at the center of the inactive galaxy G1, and hence on the M_BH-sigma_* relation at the lens redshift z_l=0.68. From detailed mass modeling, we place an upper limit on the black hole mass, M_BH < 2.1x10^{10}M_sun at 1-sigma level (<3.1x10^{10}M_sun at 3-sigma), which is consistent with black hole masses expected from the local and redshift-evolved M_BH-sigma_* relations, M_BH~10^{9}-10^{10}M_sun.
