Hi
in theory yes
but i believe you need to be in bahrain and starting work or about to before getting your bahrain work visa.

your qatar employer will be the one who will inform the authorites of your last day so that will determine when your qatar visa is cancelled

if you can move before its due to terminate then yes.
I assume you are hoping to work and then hop on plane and start work straight away in bahrain ?

my wife is moving to bahrain but her visa wont start until new job starts which overlaps her qatar visa by a day or two.

it might be worth asking your bahrain employer if they can at least prepare your work visa as to minimise the gap between finishing work in qatar and starting in bahrain,

regards