The Senegalese striker arrived from Lokomotiv Moscow on deadline day in January and made an immediate impact, scoring three times in his first four appearances.
He went on to score both goals in the victory over Crystal Palace but ended the season on a five-game scoreless streak as the Tigers slipped out of the Barclays Premier League.
N’Doye cost £3million but had a buy-out clause of £800,000 below that following relegation, a fee which the Turkish club have met.
He was absent from the squad that beat Huddersfield 2-0 on Saturday, having flown to discuss personal terms with Trabzonspor.
The 30-year-old has agreed a three-year deal.
A statement from the Tigers read: “Dame N’Doye has left the Tigers after signing a three-year deal with Turkish club @Trabzonspor.
“Everyone at @HullCity wishes Dame all the best for the future and thank him for his efforts at the club.”