No. Marriage between first cousins is prohibited in Michigan. See link provided below for a complete state listing.
Notes from FAQ Farmers:
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The state of Michigan does not allow cousins to marry.
That true but if it was your half cousin I'm sure you can marry her the law saids
"No man shall marry his mother, grandmother, daughter, granddaughter, stepmother, grandfather's wife, son's wife, grandson's wife, wife's mother, wife's grandmother, wife's daughter, wife's grand-daughter, nor his sister, brother's daughter, sister's daughter, father's sister, or mother's sister, or cousin of the first degree. "
You can't marry your cousin but you can marry your half cousin.