For the validity of qaza sawm, it is necessary to make intention before true dawn. While eating the Sehri one can make intention of qaza. But, if you did not make intention in the night and also did not wake up to eat the Sehri and kept the fast without it then the fast shall be counted as nafl and it will not be counted as qaza of the missed fast. But if you make the intention of qaza fast in the night but you did not eat the Sehri due to sleeping and kept the fast in the same state then the fast will be valid as qaza and now you need not keep the fast again.
واذا نوى الصوم للقضاء بعد طلوع الفجر لا تصح نيته عن القضاء يصير شارعاً فى التطوع . (عالمگیری: 1/197) وفى الشامي: والتبييت فى الأصل كل فعل دبر ليلاً ط . (شامي: 3/345)
Allah knows Best!
Darul Uloom Deoband