Please have a look at www.qatarpetrelocators.com as they will be able to assist you if you require it.

There is no quarantine but you will need an import permit which has to be applied for in person. You may also need an export permit and/or health certificate depending on where they are leaving from.

Are your cats travelling excess baggage or cargo as there is a difference in customs, paperwork and procedures?

As long as both your cats have valid rabies vaccines that are more then one month old there shouldn't be any problems with bringing them over.