BuddyPress.org

There’s currently no “search_terms” parameter to use in a private message loop:
https://codex.buddypress.org/developer-docs/custom-buddypress-loops/the-private-messages-loop-bp_has_message_threads/

Sounds like a good enhancement. Can you post it on Trac?
https://trac.buddypress.org/newticket

—

In the meantime, you’d have to write a custom plugin that does this.