This article explains how to save your Gmail chat history.
How to Retrieve Chat History of Your Gmail Address || How to Access Chat History in Gmail
By default, Google saves your conversations in Gmail Chats. Here's how to turn this feature on and off for individual conversations.
In a web browser, go to your Gmail inbox and select the Chat tab.
Select a chat conversation in the left sidebar, then select the person's name at the top to open a drop-down menu.
