This story is found in both the Jataka and Panchatantra traditions of India, and thanks to La Fontaine, it became part of the Aesopic tradition later in Europe: The Tortoise Flying. You can find out more in this Wikipedia article.
The illustration below is from a 15th-century translation of the story into Latin; that book, called the Directorium Humanae Vitae, became an important conduit for Indian stories that entered into the European tradition. You can see more illustrations here, and the book itself is online at the University of Munich.
SHELL-NECK, SLIM, AND GRIM
Now in the vicissitudes of time there came a twelve-year drought, which begot ideas of this nature in the two ganders: "This lake has gone dry. Let us seek another body of water. However, we must first say farewell to Shell-Neck, our dear and long-proved friend."
When they did so, the turtle said: "Why do you bid me farewell? I am a water-dweller, and here I should perish very quickly from the scant supply of water and from grief at loss of you. Therefore, if you feel any affection for me, please rescue me from the jaws of this death. Besides, as the water dries in this lake, you two suffer nothing beyond a restricted diet, while to me it means immediate death. Consider which is more serious, loss of food or loss of life."
But they replied: "We are unable to take you with us since you are a water-creature without wings."
Yet the turtle continued: "There is a possible device. Bring a stick of wood."
This they did, whereupon the turtle gripped the middle of the stick between his teeth, and said: "Now take firm hold with your bills, one on each side, fly up, and travel with even flight through the sky, until we discover another desirable body of water."
But they objected: "There is a hitch in this fine plan. If you happen to indulge in the smallest conversation, then you will lose your hold on the stick, will fall from a great height, and will be dashed to bits."
"Oh," said the turtle, "from this moment I take a vow of silence, to last as long as we are in heaven."
So they carried out the plan, but while the two ganders were painfully carrying the turtle over a neighbouring city, the people below noticed the spectacle, and there arose a confused buzz of talk as they asked: "What is this cart-like object that two birds are carrying through the atmosphere?"
Hearing this, the doomed turtle was heedless enough to ask: "What are these people chattering about?"
The moment he spoke, the poor simpleton lost his grip and fell to the ground. And persons who wanted meat cut him to bits in a moment with sharp knives.
To take advice from kindly friends
Be ever satisfied:
The stupid turtle lost his grip
Upon the stick, and died.