If a turtle is healthy, he will come up for air when he needs it. This will be more frequently if he's active and less so when he's inactive. With some variation, an active turtle can stay under water for 10-15 minutes before coming up for air. A sleeping turtle can stay under water much longer.
This assumes that there is nothing in the tank that can pin a turtle under water to prevent him from getting air when he needs it.
"You become responsible, forever, for what you have tamed." -Antoine de Saint Exupery-