Child born outside passport

I became a British citizen through naturalization before my child was born abroad, and my child is also a British citizen.

Can I apply for my child's passport within the UK, or must I apply from the country where my child was born?

You can apply from the UK, but there may be questions as per to why you are doing so (how would you bring your child to the UK without a visa for example). You can apply from abroad, but be mindful that such applications can take longer to process.