He doesn’t tell her his name at first, but he absolutely does.
A little later in the episode she reacted to him saying his name, and that’s what led her to acquiesce and bring him to the Watcher.
I'm less certain now, but why did the producer go out on Twitter and apologize for Guinan not recognizing Picard, and explain that that they didn't meet in 1893?
https://twitter.com/TerryMatalas/status/1507019504663945219