Where would the quarter moon be in relation to the earth?

When the moon is in the same direction of the sun, it is called a new moon. A week after the new moon, when the moon has completed about a quarter of it's turn around Earth, this is called the first quarter and about three weeks after the new moon and a week after the full moon, it is called the last quarter.

If you visualise the moon turning around the Earth and imagine that the direction of which the sun is shining is called the "front' of the Earth, the the quarter moon would be at the 'sides' of the Earth and the full moon is at the 'back'.