Do Mormons believe that Michael the archangel was Adam from the bible?

Yes. This is not something that is commonly taught or preached during Mormon services, but it is a Mormon belief that was taught by Joseph Smith. During the Temple Endowment ceremony, Mormon adults are taught that Michael became Adam, the first man. In the Church-published "Guide to the Scriptures" it says "Michael is the name by which Adam was known in the premortal life. He is called the Archangel. In Hebrew the name means 'Who is like God'. Michael or Adam is the archangel or chief angel."

For more information, search for the sermon "Adam, the Archangel" by Elder Mark E. Peterson given in October 1980. Video, audio, and text of the sermon can be found on lds.org.