Does this sound natural ?
A: Are you good at English ?
K: Sure ! I'm really really confident in English.
English Language & Usage Stack Exchange is a question and answer site for linguists, etymologists, and serious English language enthusiasts. It only takes a minute to sign up.Sign up to join this community
It's not wrong, but it's not necessarily a natural idiom that a native speaker would use. You might say "I'm proficient in English, which is a bit more formal, or I'm great at English, which is more informal, but sounds more confident. If you wanted to use the actual word confident, you'd need something like I'm confident in my English skills --which actually doesn't sound that confident at all.
Your phrase, as it stands, technically means that you are confident when you speak English, but not necessarily because you speak English well.
No, it doesn't sound natural. Usually one would respond with the same terms as being asked.
The asking party is asking about the ability:
Are you good at thing?
The proper response is, if true:
Yes, I am [very or really] good at thing.
No, I am not [very or really] good at thing.
More information on really vs very: ELU, Learner's Dictionary. And if you're really interested in a long treatise on really, Between epistemic modality and degree: the case of really (pdf) by Carita Paradis