No. In order for a quadrilateral to be a trapezoid, it must have exactly one pair of parallel sides. A right trapezoid, therefore, has exactly one pair of right angles. However a square would be a right parallelogram (which is the current definition of a rectangle).
Is this a trapezoid?
A trapezoid is a quadrilateral with one pair of opposite sides parallel. It can have right angles (a right trapezoid), and it can have congruent sides (isosceles), but those are not required.
What is a trapezoid shape?
In summary, a quadrilateral is any four-sided shape. A parallelogram is a four-sided shape with opposite sides that are both parallel and equal in length. The area of a parallelogram is the base times the height. A trapezoid is a four-sided shape with at least one set of parallel sides.