S.H.
Is she eating normally and adequately, during the daytime and when she is not sleeping?
That would be my other concern. You want to make sure an only 4 month old baby is getting adequate intake/caloric intake/nutritional intake. Daily.
If your baby is fine in growth/weight gain/development, then that is probably fine. And maybe that is just the way your baby is. BUT you should ask your Pediatrician.
According to my Pediatrician: for the 1st year of life, a baby must be fed ON demand, and breastmilk/Formula is a baby's PRIMARY source of nutrition, NOT solids, NOT other liquids. Keeping in mind that for the 1st year of life, feedings ALSO provides crucial "building block" nutrition for a developing baby.
I know of one baby, that was a sort of "lazy" feeder. I don't know how else to describe it... but even if she was hungry, she didn't indicate it and she just went with whatever pattern her Mom did to her. BUT, she was a real smallish baby, not real alert, and wasn't gaining much weight. So, there is that other side to the issue too. NOT saying this applies to you.
Just ask your Pediatrician. Every baby is different. I think the overall concern, would be that on a DAILY basis, that baby is getting the proper amounts of intake. Versus "when" she got that intake amount.
BUT I would certainly feed ON demand, when a baby is having a growth-spurt... because at these times, a baby NEEDS increased intake because they are growing so much and their intake needs to keep pace WITH their growth.
To me, a baby will not become entrenched in the 'habit' of being fed every single time they wake. AND by the time a "baby" grows older and becomes a Toddler, they will not have this habit... it naturally changes per the age of the child and their natural needs for nutritional intake. Just because you feed a baby when they wake at night... does NOT mean that you have to do this for the rest of their life... nor will the "child" expect that... a baby changes into a Toddler and then an older child, and over the course of their changes/growth, the expectations of that disappears. They are not a baby forever. And they will NOT expect to "eat" every time they wake. My kids, are actually NOT hungry "as soon as" they wake up....but I fed on demand when they were babies and whenever they woke up during the night. To me, the "logic" that a baby will 'get used to' being fed each time they wake and that it is a 'bad' thing, is over-rated and not true. And, just because a baby wakes it does not mean you stick a nipple in their mouth either... it depends on the CUES that a baby displays... if a baby IS hungry, then a Mom will know that. There are hunger cues, and there are just noises a baby makes or transient wakings during the night, or just soothing that they need.
But yes, each baby is different...
All the best,
Susan