The 28-year-old is heading to France for an undisclosed fee after two years on Teesside, although the the length of his contract has yet to be revealed.
Fabio wrote on Twitter: "I want to say a big thank you to everyone @Boro Staff, players, most of all the fans.
"I tried to do my best every day and give my all, I hope you can return to the Premier League very soon.
"Thank you, you will always be in my heart. Up the Boro #UTB."