Short answer: QMB applies to services furnished after the end of the month in which the state determines the person eligible, so in practice it starts the first day of the next month. It is not retroactive to earlier months.
Federal Medicaid law (42 U.S.C. 1396a(e)(8)) says a QMB determination applies to services furnished after the end of the month in which it is first made. A person approved on March 10 is covered for Medicare cost-sharing beginning April 1. The same provision treats a determination as good for 12 months for federal funding purposes; a state may recheck more often, but not more than once every 6 months.
Practical point: bills for care before the QMB start date are not covered by the QMB billing protection. File the application as soon as a person looks eligible rather than waiting for a large bill.