Notions of linear sufficiency and quadratic sufficiency are of interest to some authors. In this paper, the problem of nonnegative quadratic estimation for β'Hβ+ hσ2 is discussed in a general linear model and its transformed model. The notion of quadratic sufficiency is considered in the sense of generality, and the corresponding necessary and sufficient conditions for the transformation to be quadratically sufficient are investigated. As a direct consequence, the result on (ordinary) quadratic sufficiency is obtained. In addition, we pose a practical problem and extend a special situation to the multivariate case. Moreover, a simulated example is conducted, and applications to a model with compound symmetric covariance matrix are given. Finally,we derive a remark which indicates that our main results could be extended further to the quasi-normal case.