Is it permissible to complete a fard (obligatory) fast after intending to fast subsequent to the Fajr athan, and would this fast be counted?

If you are observing an obligatory fast, such as in Ramadan, and you did not intend to fast until after Fajr prayer, then your fast is invalid and must be made up, according to the majority of scholars. However, the Hanafis permit delaying the intention until the forenoon (Duha) time. But if your fast is a voluntary one, then you may intend to fast after Fajr and complete the day, provided that no act that breaks the fast has occurred.