A trapezoid is a quadrilateral because it has 4 sides. A quadrilateral is not always a trapezoid because a quadrilateral can be any shape as long as it has four sides.
A quadrilateral can be any shape with 4 sides, for example parallelograms, rhombi, squares, kites, and also trapezoids. Since a quadrilateral can be any of these that I just listed, a quadrilateral is not always a trapezoid.
