Not really. The "Batman: Under the Hood" comics came out in 2005. "Under the Hood" was the name Hollis Mason's fictitious biography in Alan Moore's "Watchmen", which pre-dates that by almost two decades. I'm sure there have been countless references to "Under the Hood" in superhero pulps through the years, and probably in older literature as well, so trying to pin down the exact first appearance of the phrase is probably a fool's errand, but rest assured it did not begin with Batman.