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I've watched all the Godzilla movies multiple times and the only one I recall any egg laying in was the American one from 1998. He adopted had adopted one in the Toho movies but didn't lay it.
I felt like I remember it happening more times 🤔
https://www.youtube.com/watch?v=hy4YFaCFGoM This video covers the various eggs in the Godzilla franchise. Most of them are Mothra but it has the ones that from Son of Godzilla, Godzilla vs. Mechagodzilla II, and the Roland Emmerich movie, which lines up with with I remembered. The origins of the Son of Godzilla and Mechagodzilla II ones are not stated but it's never implied that he laid them.
After a minute of research I'm inclined to believe Godzilla egg-laying only happened in Roland Emmerich's 1998 film.
Here is some contemporary reporting about it: https://www.chicagotribune.com/1998/05/19/godzilla-lays-an-egg-does-this-surprise-you/
Big, buff and bodacious, he’s so cool he can even reproduce himself–or herself. Turns out, Godzilla’s a hermaphrodite.
Consistent with the mythology, this giant lizard is a mutant by-product of nuclear radiation. As the only member of its species to have survived a bomb test in French Polynesia, Godzilla must assume male and female reproductive functions to maintain the lineage.
Why Godzilla feels compelled to travel all the way to Manhattan to lay its eggs is a mystery not clearly explained in the script, but, like any Sinatra fan, the monster probably thought, “If I can make it there, I’ll make it anywhere.” So, it was off to New York, New York, where–like the Knicks–the creature lays a lot of eggs in Madison Square Garden.
see also: https://fictionhorizon.com/how-does-godzilla-reproduce/