Siri and Susebron Talk about How the Next God King is Created
Siri's impulse here—that the next God King might not really be the son of the current one—is a good one. She's actually right, though there are a lot of other things in this conversation she's wrong about.
It is possible for a Returned to have a child. Vo, the First Returned, did it. The God King isn't special in that he can do it; any of the Returned could, but it requires some special knowledge that—I'm afraid—I'll have to keep secret until the sequel. Suffice it to say that the priests know how it is done.
The problem is, they aren't always able to get this to work. Sometimes, they have to do what Siri guessed—replace the God King with an infant Returned. Infant Returns happen very infrequently. It's more rare than an adult Returning, so there is something sound to the Hallandren reasoning that you have to do something heroic in order to Return. (That's not true, but it is more sound a doctrine than Siri thinks it is.)
The God King's priests take an infant Returning as a sign that it's time to change God Kings. At that point, they choose a wife for the God King and hope that she'll be able to conceive the next God King. They'd much, much rather that the God King be the literal child of the previous God King. (Susebron wasn't, however. And his mother was indeed his mother, a poor merchant's wife from far northern Hallandren.)
Now, an infant has indeed Returned. The priests see this as a major vindication of their faith, as they made the wedding contract with Idris twenty years ago and now, just when the marriage was to happen, an infant Returned. The problem is, now they've got to push Siri to get pregnant, because they're on a deadline. They don't want to have to replace the God King with this infant; they'd rather use his own child. Hence the push for her to have a child.
But if she doesn't, they'll go with plan B. Note that there's not, in fact, any danger to her either way, no matter what Bluefingers says. She and Susebron, following the change in power, would have been taken to one of the isles in the middle of the Inner Sea and kept in a lavish lifestyle as long as they lived.