First, remember that 'acceleration' is any change in the object's speed or direction. So the object is accelerating if it's speeding up, or slowing down, or moving in a curve.
If it's speeding up or slowing down, then the graph of its speed vs. time is anything different from a horizontal line.
If it's moving in a curve but maintaining a constant speed, then the speed vs. time graph doesn't reveal it, although the object is technically still accelerating.