We investigate the Korn first inequality for quadrilateral nonconforming finite elements of first order approximation properties and clarify the dependence of the constant in this inequality on the discretization parameter $h$. Then we use the nonconforming elements for approximating the velocity in a discretization of the Stokes equations with boundary conditions involving surface forces and, using the result on the Korn inequality, we prove error estimates which are optimal for the pressure and suboptimal for the velocity.