Hello, I am confused on how to make a new section in my hatchery, I understand how to do it in stables by purchasing it with 500 gems but I can't find that same button or option in the hatchery, help?

As far as I know, the hatchery expands automatically, so you don't have to buy anything.

When you get enough eggs to fill one page, the next egg (or hatchling) will automatically create another page.

There is no limit to space in the hatchery, as stated above; basically, you can have an infinite number of eggs/hatchlings/adolescents! Just be warned that if you don't have enough space for the pets in your hatchery, they'll be released upon becoming adults.