I agree with Vince.:
It sounds like your file geodatabase has been corrupted for geodatabase behavior. QGIS uses one of two potential libraries, neither of which is geodatabase aware. Please edit the question to specify the version of QGIS and whether you are using the Esri or open access library. If ArcGIS says the geodatabase is corrupt, then it's in the best position to know; you may need make use of your dataset backup.
The GDB sounds corrupt. So long as you can still access the data within from QGIS I would export everything to a new GDB and delete the corrupt entity before the problem spreads.
