(Oxford English Dictionary)
MOVING DAY, n.
1. The day of a move to a new residence; (U.S.) the day on which leases normally expire in a particular city or region (in New York, May 1), frequently associated with household removals.
1832 J. F. WATSON Hist. Tales N.Y. 123 'Moving day' was, as now, the first of May. 1852 W. B. JERROLD & W. H. WILLS Househ. Words 2 Oct. 62/2 Being 'moving' day, it rains. 1855 Knickerbocker 45 585 In the southern part of New-Jersey, one who rents or purchases a house or farm usually takes possession of the same on the twenty-fifth day of the present month [sc. March], which is therefore denominated 'moving-day'. 1947 Pasadena (Calif.) Star-News 9 Sept. 16/6 Moving days are ahead for several county departments in the Hall of Records and Hall of Justice, Los Angeles. 1973 National Observer (U.S.) 3 Feb. 7/1 When our moving men finally showed up ('removals' in their business, whether you are coming or going, and on moving day we hardly knew which we were) we received another language lesson. 1992 Canad. Living Aug. 13/2 Don't have a garage sale too close to moving day.
I can provide a slightly earlier citation.
8 May 1830, Workingman's Advocate (NY), pg. 3:
We dislike, above every thing, (except "moving day," vulgarly called "May day,") a milk and water politician.