No, the full Moon is never overhead at noon.
The moon will be full around midnight, rising at around 6 p.m. and setting at about 6 a.m. The lit side of the Moon faces the Sun and the other side faces the Earth, signifying a new moon, if the Sun, Moon, and Earth are in a straight line (in that order).
As a result, we can see both the moon's dazzling and dark faces. Because the full Moon happens when it is in the opposite position from the Sun in the sky, we cannot view one at noon. At midnight, the full moon is at its highest point in the sky.
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