Is it only if the child is the MOTHER’s firstborn son?
Yes. How many children the father has is irrelevant.
I know a previously never married man who married a divorcee with two daughters. His wife gave birth to a son about two years ago — and there was no pidyon, simply because it was not her first child. The fact that he never previously had children does not matter.
OTOH, a bechor for the purpose of inheriting a double-share depends on the father, not the mother*.
The Wolf
* Although if the bechor came into the world via Cesarean, then he does not qualify for the double share.