No, MGM is now a seperate corporation. Back in the 1980's MGM was bad off financially, and the company was pretty much bankrupt. Their movie studio backlot facilities were sold to Sony, the production company to private investors, and other peices were sold off.
Ted Turner, who was looking for programming for his TBS Superstation, purchased the rights to broadcast the HUGE MGM film library (largest in the world with over 4000 titles). For a time, his TBS Superstation (and later TNT and Turner Classic Movies) were the only channels showing MGM films that were made prior to 1984(?) Ted Turner sold his holdings to Time Warner, so the movie broadcast rights became part of the Time Warner family, and they were just put under the Warner Brothers arm of the company.
So to answer your question, no MGM is not part of Time Warner (at least not yet)
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