It's more about your prep than the age of the child. My friend had some taken in the hospital that were wonderful. so if a day old baby can do it, anyone can.
It would probably go smoother if you have a helper, that way you can talk to the photographer and handle picking picks and paying at the end while someone holds the baby.
The child needs to be fed and well rested. bring things they love like a cool truckor a stuffed animal. if the actual time in front of the camera is more than a half hour and you aren't gettting good picts, then call it a loss and be done.
If it's a stranger anxiety thing, when you make the appointment ask for the most low key photographer. if your child is exuberant and you think htey will want to run and run ask for a very vibrant photographer.
They can do all sorts of shots that show you holding the baby's hands or a toddler wrapped on your leg playing peek a boo. if you don't like your face in shots or have a few pounds on the tummy those parts don't have to show.
Why don't you google the studio you want to use, they usually have a gallery so you can see what some of the picts look like. I use JC Penny's most of the time, we don't have a target photo, the have some you can look at