I’ve seen this go both ways with Qatar.
They can extend vouchers, especially for genuine life events like a newborn, but you usually only get traction if you contact them directly and escalate a bit. It’s not something the system does on its own. Phone or chat, explain the situation clearly, mention the January birth, and ask for a one time goodwill extension. Having the birth certificate now helps a lot.
As for booking now: yes, you can usually book tickets for the adults and existing child using the vouchers and then add an infant in lap later. Infant tickets are normally added after booking anyway and don’t require their own seat. You typically don’t need the infant’s passport number at the time of the adult booking, just before ticketing the infant.
Passport number timing is the bigger risk, but airlines don’t always require passport details at booking, only at check in or later in “manage booking”. If the site forces it, an agent booking over the phone often has more flexibility.
Biggest advice: don’t let the vouchers expire silently. Call Qatar now, document everything, and ask specifically for an extension or for permission to book now and add the infant later. Qatar is generally reasonable when you get the right agent.
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