From time to time people do put up posts at QL with regards to handing over their pets when such people are leaving Qatar for good. Suggest, you be a bit patient and wait for such ads. Sooner or later, you will find a dog of your choice.
As for getting one from a neighboring country, I am not sure if you could do that. It may involve a lot of hassle. Possibly some QLer may be in a position to guide you on this.