The Qatar Airways pet policy is as follows:
Pets are not accepted in the cabin.
Service dog(s) can be transported in the cabin free of charge when travelling on specific routes.
Qatar Airways allows you to carry one falcon in the Economy Class cabin of an aircraft, and a maximum of six falcons are permitted within the Economy Class cabin of any one aircraft.
Qatar Airways accepts domesticated dogs, cats, and birds as checked baggage, either on your flight or on a separate one.
Qatar Airways does not permit transport of certain breeds of dogs and cats as these animals are normally susceptible to increased risk of heat stroke when exposed to stress or temperatures above 21°C (69.8°F).
Taking a pet to a foreign country involves preparation and planning far in advance of your departure date. Requirements for pet travel to a foreign country are set by the destination country. If you are you traveling to another country and bringing along your pet you’ll need to meet the animal health requirements of the country you are visiting.
More information on the Qatar Airways pet policy