I'll take a shot at this riddle. I am not quite as turtle savy as I am crocodilian savy. However, they are both similar in several ways. If this was a crocodilian, I would first take a look at the water temps. What are this water temps like? If the water is warm enough that he does not need to thermoregulate, then he usually wont. Also, what types of stressors are around him? If you are constantly in the room, he might feel stressed about basking for some odd reason. Therefore, he might only be coming out to bask when the entire area is quiet and there are no other life-forms that can bother him near by. I would take a look at water temps, compared to basking temps. If he would be cooler by coming out of the water, I am sure he would prefer to stay in the water. Vice versa if the water was too cool.
I could be incorrect on this assumption however, but crocodilians and turtles are very similar in many aspects. This is how I would typically treat this issue if this were my alligators.
Hope this helps in some small way!
Best Wishes,
Chris