A. It is a common misunderstanding that infants do not have sinuses.
In fact, babies are born with sinuses, but they are just not very well developed. Newborns have very small maxillary (under your cheeks) and ethmoid sinuses (between your eyes), but they can't be seen on regular xrays until a child is 1-2 years old.
The frontal sinuses (which often contribute to typical adult sinus headaches) and sphenoid sinuses don't begin to develop until a child's second year and can't be seen on an xray until the child is 5-6 years old. The sinuses continue to grow until your child is a teenager.
How long had your child been sick?
If it had been for more than 10-14 days, then she may very well have a sinus infection. But whether or not she could have a sinus infection is controversial. Most likely, even if she has a green or yellow runny nose for several weeks, instead of a real sinus infection, it would be more appropriate to just call it purulent rhinitis. If it was lingering and getting worse, you would still treat her with antibiotics though.
If she has only been sick for 5-7 days, then she may just have a cold.