To my knowledge Mehmed (or Mehmet) is the "Turkified" version of Mohammad (as in the prophet). "Mahmud", the name that you suggest as the root for "Mehmed", is commonly used in Turkish as "Mahmut". And actually Mehmed sounds closer to Muhammed (Turkish version of Mohammed) than to Mahmud or Mahmut.
― Anonymous User 4/4/2005
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https://en.wiktionary.org/wiki/Mehmed
https://en.wiktionary.org/wiki/%D0%9C%D0%B5%D1%85%D0%BC%D0%B5%D0%B4 [noted -ed]