No, actually too many Nos. I assume that your UAE residency will be cancelled before leaving Dubai, that won't happen till the maid's RP is canceled as well. So she would no longer be able to enter Qatar "with you" even though you can get a visa at the airport. I took my maid to Oman during our vacation, we got airport visa for all, but she was under our sponsorship then.

You can arrange with one of the maid agencies here to re-recruit her once your Qatar RP is completed.