Lea Michele's Boyfriend -- Check Out My Hot Ex-Gigolo ... Coach

Lea Michele stepped out publicly for the first time with her new guy Matthew Paetz ... but the photos look about as genuine as Matthew's claim that he wasn't a gigolo.

The pics show Lea and Matthew, guns out, on a leisurely hike -- and look like classic set up shots pre-planned with photo agencies. They're even wearing matching sneakers.

Our take -- Lea is screaming to the world she doesn't care about Paetz's past -- remember he claims he was only coaching gigolos, not doing the gigoloing himself.