There are many ways to resolve this:

Ist is try to shrink this DB and then take a fresh backup which can be restored at the destination(QA) Server. [but this can result in fragmentation]

2nd is if the DB has only 1GB of data you should create a fresh DB with initial size 1 GB and do enable the autogrowth but only by 10% and then shift all your data to that DB and then take a fresh backup and restore it.