Was Moab the same as Midian in the Old testament?

No, Midian was the son of Abraham and Keturah (his second wife). Midian moves east of Canaan. Midian (the place) is hard to pinpoint, but it may have been in present day Jordan at one time, later moving southward into Arabia on the east bank of the Red Sea, possibly shifting even further into the Sinai Peninsula later on.

Moab on the other hand, was the son of Abraham's nephew Lot. Therefore, Midian was Abraham's son, and Moab was Abraham's nephew. So Midian and Moab were cousins. Lot got drunk and slept with his two daughters. Moab and Ben-ammi were the results of these incestuous relationships. Moab and Ben-ammi became ancestors of two nations that later fought against the nation of Israel - Moab and Ammon, both in present day Jordan.

Hope this helps.