In the process of grey prediction modeling, the accumulation of original non-negative sequences can enhance the inherent regularity of data and the smoothness of sequences. The estimated method of background values and derivative values greatly affects the prediction accuracy and adaptability of the model. On the basis of the traditional GM(2,1) model, the GM(2,1) model with the fractional order accumulation is obtained by introducing a fractional order operator, which can be written as GM ${}^r$(2,1) with r-AGO. In this work, we estimated the background values and derivative values based on a $C^1$ convexity-preserving rational quadratic interpolation spline, and thereby established a new GM ${}^2$(2,1) model. Numerical examples showed that the new GM ${}^2$(2,1) model had better prediction quality of data than the classical GM(2,1) and GM ${}^2$(2,1) model and improved the precision of prediction in practice.