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We present a spectroscopic analysis of the metallicities, ages, and alpha-elements of the globular clusters (GCs) in the giant elliptical galaxy (gE) NGC 4636 in the Virgo Cluster. Line indices of the GCs are measured from the integrated spectra obtained with Faint Object Camera and Spectrograph on the Subaru 8.2 m Telescope. We derive [Fe/H] values of 59 GCs based on the Brodie & Huchra method, and [Z/H], age, and [α/Fe] values of 33 GCs from the comparison of the Lick line indices with single stellar population models. The metallicity distribution of NGC 4636 GCs shows a hint of a bimodality with two peaks at [Fe/H] = -1.23(σ = 0.32) and -0.35(σ = 0.19). The age spread is large from 2 Gyr to 15 Gyr and the fraction of young GCs with age <5 Gyr is about 27%. The [α/Fe] of the GCs shows a broad distribution with a mean value [α/Fe] ~0.14 dex. The dependence of these chemical properties on the galactocentric radius is weak. We also derive the metallicities, ages, and [α/Fe] values for the GCs in other nearby gEs (M87, M49, M60, NGC 5128, NGC 1399, and NGC 1407) from the line index data in the literature using the same methods as used for NGC 4636 GCs. The metallicity distribution of GCs in the combined sample of seven gEs including NGC 4636 is found to be bimodal, supported by the KMM test with a significance level of >99.9%. All these gEs harbor some young GCs with ages less than 5 Gyr. The mean age of the metal-rich GCs ([Fe/H] >-0.9) is about 3 Gyr younger than that of the metal-poor GCs. The mean value of [α/Fe] of the gE GCs is smaller than that of the Milky Way GCs. We discuss these results in the context of GC formation in gEs.