For quite some time now Apple has been very strongly urging users to use Two Factor Authentication instead of security questions. --> https://support.apple.com/HT204915 "With two-factor authentication, you don't need to remember any security questions. We verify your identity exclusively using your password and verification codes sent to your trusted devices and phone numbers."
If you have a new Apple ID and are in a country which does use Two Factor Authentication, you will not have the option to use security questions.
There are a few countries where Two Factor Authentication is not available. Are you in one of those?
Availability of two-factor authentication for Apple ID - https://support.apple.com/HT205075
- List of countries where available