  We introduce simple quadrature rules for the family of nonparametric
nonconforming quadrilateral element with four degrees of freedom. Our
quadrature rules are motivated by the work of Meng {\it et al.}
\cite{meng2018new}. First, we introduce a family of MVP (Mean Value
Property)-preserving four DOFs nonconforming elements on the intermediate
reference domain introduced by Meng {\it et al.}. Then we design two--points
and three--points quadrature rules on the intermediate reference domain. Under
the assumption on equal quadrature weights, the deviation from the
quadrilateral center of the Gauss points for the two points and three points
rules assumes the same quadratic polynomials with constant terms modified.
Thus, the two--points rule and three--points rule are constructed at one
stroke. The quadrature rules are asymptotically optimal as the mesh size is
sufficiently small. Several numerical experiments are carried out, which show
efficiency and convergence properties of the new quadrature rules.
