Yes. Qatar Airways allows you to book your flights with infants on qatarairways.com. At the time of departure for each flight during the course of the journey, infants must be older than 7 days and younger than 2 years old.
If the infant turns 2 during the course of your journey, you will need to book a child fare ticket for the entire journey, if you are booking online.
Alternatively, you may visit your nearest Qatar Airways ticketing office or your preferred travel agent to book his or her ticket. You may either purchase a child fare ticket for the entire journey, or purchase two separate tickets i.e. an infant fare ticket for the commencement of the journey and a child fare ticket for the return/onward journey.