Yes, most likely, according to Gandalf
Gandalf does an excellent job of describing Treebeard in the chapter “The White Rider”
Treebeard is Fangorn, the guardian of the forest; he is the oldest of the Ents, the oldest living thing that still walks beneath the Sun upon this Middle-earth.
The Two Towers, Book III, Chapter 5: The White Rider
We know that ents were created before the first age by the pleading of Yavanna
then the thought of Yavanna will awake also, and it will summon spirits from afar, and they will go among the kelvar and the olvar, and some will dwell therein, and be held in reverence, and their just anger shall be feared.
The Silmarillion, Chapter 2: Of Aulë and Yavanna
While it is unknown when, Treebeard also speaks about the Elves waking the trees and teaching the Ents to talk.
Elves began it, of course, waking trees up and teaching them to speak and learning their tree-talk...
The Two Towers, Book III, Chapter 4: Treebeard
Finally, Legolas and Aragorn describe Fangorn (the forest, although based on Gandalf’s quote also the Ent) as older than Men, while the Elves walked under the light of the Trees alone.
... for Fangorn is old, old even as the Elves would reckon it.'
'Yes, it is old,' said Aragorn, 'as old as the forest by the Barrow-downs, and it is far greater. Elrond says that the two are akin, the last strongholds of the mighty woods of the Elder Days, in which the Firstborn roamed while Men still slept...'
The Two Towers, Book III, Chapter 2: The Riders of Rohan
Based on the first quote, and all the supplementary information, the Ents are likely very old and Treebeard being the oldest has lived to a great age.