If you record the chat conversations you can use the API to get all the messages associated with a Conversation ID:
/api/v2/conversations/{conversationId}/recordings
However, you would need to do a lot of cleanup to get it in the format you want.
There is an Idea in the Ideas Lab that you can vote for:
https://purecloud.ideas.aha.io/ideas/CLDIG-I-299------------------------------
George Ganahl GCP (Genesys Cloud), ICCE
Principal Technology Consultant
Genesys
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