"Good", when describing a person, is normally used to refer to their character, rather than to their health. Hence "I'm good" would seem to be saying "I'm a good person", which isn't what's required in this context.
"Well" refers to health, rather than to character. Hence "I'm well" is the grammatically correct response here.
"Good" would only be appropriate if the questioner specifically mentioned the other person's health. (e.g. "How's your health these days?"). Then "Good, thank you" would be an appropriate response.