When trying to turn off "Conversation History" in Hangouts on desktop from here: https://hangouts.google.com/, there is no option for this as stated in the help doc. Others have reported this last year to the Google Support thread below (which is closed, wtf?), but even the screenshot there of where it should be is different from what looks like a new layout. The help docs instruct as follows:
1 On your computer, go to Hangouts at hangouts.google.com or in Gmail.
2 Open a conversation.
3 At the top, click Settings Settings.
4 Check or uncheck "Conversation history."
"Delete or turn off your message history"
https://support.google.com/hangouts/answer/3112001?hl=en&co=GENIE.Platform=Desktop
"Option to turn conversation history on or off isn't in the settings menu on any conversation."
https://support.google.com/hangouts/thread/4711330?hl=en
As suggested below to create another Google Support request:
"Still a Problem: Option to turn conversation history on or off isn't in the settings menu on any con"
https://support.google.com/hangouts/thread/40049352?hl=en
And my question on why Google locks questions still needing a reply:
"Why do you let robots lock human questions needing an answer?"
https://support.google.com/accounts/thread/40069094?hl=en
Best Answer
If the setting is missing and you already tried basic troubleshooting the submit your feedback to Google by using Google Feedback
From Hangouts,
It also could be a good idea to start a new thread in the Hangouts Help Forum / Community. By the way on Google Products Help Forums inactive threads are automatically closed after some time.
Basic troubleshooting