Ensure you are compliant with the standards
In order for the NFT to show up in the API it needs to be compliant with ERC721 or ERC1155 standards.
Both of these standards require the implementation of supportsInterface method from ERC165 standard.
If you use OpenZeppelin contracts this is done automatically for you.
A way to double check this is to open your contract in Etherscan and ensure it says ERC721 or ERC1155 on your contract page.
A contract like this will show up in the NFT API.
A contract like this won't show up in the NFT API.
You can also use the Resync NFT Contract and Resync metadata endpoint to refresh the metadata of the contract or NFT.