In this paper, we proposed a novel binary classification framework named adaptive robust AdaBoost-based kernel-free quadratic surface support vector machine with Universum data (A-R-U-SQSSVM). First, we developed R-U-SQSSVM by integrating the capped $L_{2,p}$-norm distance metric and the generalized Welsch adaptive loss function to improve the model's robustness and adaptability. Furthermore, we introduced Universum data points into R-U-SQSSVM to enhance the model's generalization performance by incorporating valuable prior knowledge for the classifier. Additionally, we utilized R-U-SQSSVM as a weak classifier and embedded the AdaBoost algorithm within it to obtain a strong classifier, A-R-U-SQSSVM. To effectively solve our model, we transformed it into a quadratic programming problem using the half-quadratic (HQ) optimization algorithm and concave duality. This transformed problem can be solved using convex optimization methods, such as the sequential minimal optimization (SMO) algorithm. Experimental results on University of California, Irvine (UCI) datasets demonstrated the superior classification performance of our method. In large datasets, A-R-U-SQSSVM was hundreds or even a thousand times faster than traditional capped twin support vector machine (CTSVM), SQSSVM.