So... is Tantei Gakuen Q finished or not?
NO, it is not finished.
Why?
The anime finished before the manga:
Anime finished: March 20, 2004
Manga finished: August 22, 2007
MangaUpdates contains information about the anime adaptation and indicates that it is incomplete.
Watching Order (by release date)
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Synopsis
Tantei Gakuen Q follows Kyuu, an ordinary boy with extraordinary deductive skills, who aspires to become a detective.
He discovers the prestigious Dan Detective School (DDS), a unique institution that trains aspiring detectives.
Kyuu enrolls in DDS and teams up with a diverse group of talented classmates: Megu, who possesses photographic memory; Kinta, a martial arts expert; Kazuma, a brilliant programmer; and Ryuu, an enigmatic individual.
Together, this team of young detectives-in-training uses their distinct abilities to solve complex and cleverly designed criminal cases, always striving to uncover the truth behind each mystery.