No one seems to be able to answer that question, though a lot of people ask it.
In the first season it's Agent Gibbs (played by series star Mark Harmon) who's dropping all the movie lines, while Agent Tony DiNozzo (played by Michael Weatherly) listens cluelessly. In later seasons, though, DiNozzo is constantly dropping lines and references to classic films, mostly to erstwhile friend and partner Ziva.
All that's left is to guess that it was a character retooling that came along later, as he had definitely become the film-quoting agent we now know by Season 3 (the season in which Ziva was introduced).
Regardless, this results in a few continuity problems that can prove aggravating for devoted "NCIS" fans like yourself. Though as you point out, in the Season 1 episode "Minimum Security," he'd said he'd never seen "The Maltese Falcon," but by the sixth season (specifically the episode "Capitol Offense") he was quoting from the movie and knew what kind of gun Bogart carried.
It would even be tough to argue that he'd devoted himself to the study of classic cinema in the years between the first and third seasons as some in the active web-based fan community have suggested, since in the Season 8 episode "Baltimore," which flashes back to his pre-NCIS days as a Baltimore cop, he makes a casual reference to the 1955 classic "Night of the Hunter."
DiNozzo's movie knowledge has become an endearing part of the character for many, as proven by the number of pages online devoted to recounting his many references, and even the multiple YouTube mash-up videos that string dozens of them together.
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