It can include linguistic and non-linguistic elements.
Example
ICC could include understanding how gestures and the distance between speakers vary from culture to culture.
In the classroom
Ways to develop this competence include learners producing a written or online guide to their own country and culture for visitors, reading and discussing guides written by visitors, researching aspects of a target culture in various media including cinema, literature and television, giving presentations on aspects of the target culture, and exploiting the teacher's own expertise of their own culture.
Further links:
https://www.teachingenglish.org.uk/article/making-culture-happen-english-language-classroom
https://www.teachingenglish.org.uk/article/storytelling-celebrate-cultural-diversity
https://www.teachingenglish.org.uk/article/culture-fifth-language-skill
Learning a language is very complex and should be done by following some secure steps, that’s why we have language levels, usually we don’t step to the next level until we don't reach acceptable result in the current level. It’s brilliant you want to include culture in teaching a language but I think it’s very important to know when the right moment for that is, it’s important in public schools but also in private esl tutoring.