How would you drive here? You, as a female, are not allowed to drive in Saudi Arabia. I don't know where you're coming from but if, for example, you're coming from UAE - you would need the proper documents issued by the UAE authorities/vets, probably documents allowing you to import/transit an animal via Saudi (if a man is doing the driving and if you are married to him, his sister, or his daughter or his mother) and then more docs to allow the dog to enter Qatar.

If you're coming from further afield and have more countries to pass through, it gets even more complicated on the document side. I suspect nearly all of these countries would require a vet inspection by their vet before allowing the dog entry/transit through their country