How to achieve certificate based authentication with AWS Cognito?

AWS Cognito as an authentication method for my cloud application. . I didn't find any forum addressing this

I believe the the documentation is quite comprehensive. You can use AWS Cognito simple as an OAuth 2.0 IdP

AWS Cognito doesn't use public key certificates?

No, it doesn't

how to achieve certificate-based authentication with AWS Cognito?

You will need to federated the authentication with a provider, which does.

AWS Cognito can be federated using SAML with an external IdP (it is considered as an enterprise feature so it costs something, check the current pricing, I recall it was around $0.5 / MAU (monthly authenticated user) )

I am not sure if there is a cloud provider directly supporting X509 based user authentication, you may setup your own SAML based identity server (there are a few product supporting X509 authentication, from small and easy SimpleSAMLphp to full identity servers such as WSO2 Identitiy Server, ForgeRock IAM or your corporate AD)