from feedback that i got from nurse-friends with Hamad Hospital, finding work as nurse on government hospital(while coming to qatar on a tourist visa) is a very slim possibility.

I have no idea what is your chance to get a job as nurse on private hospitals, probably some private hospitals do local hire. try to get more info.

Get help from friends or relatives already working with Hamad Hospital and they might have some info on hiring. I heard hiring for filipino nurses is normally done through an assigned employment agency in the philippines.