Because of the limitations of the infrared imaging principle and the properties of infrared imaging systems, infrared images have some drawbacks, including a lack of details, indistinct edges, and a large amount of salt-and-pepper noise. Traditionally, the total variation (TV) regularization method with L1 norm is used for image deblurring in preserving edges and removing salt-and-pepper noise. However, the TV-based solutions usually have some staircase effects. To improve the sparse characteristics of the image while maintaining the image edges and weakening staircase artifacts, we propose a method that uses the Lp quasinorm instead of the L1 norm and for infrared image deblurring with an overlapping group sparse TV method. The Lp quasinorm introduces another degree of freedom, better describes image sparsity characteristics, and improves image restoration. Furthermore, we adopt the accelerated alternating direction method of multipliers and fast Fourier transform theory in the proposed method to improve the efficiency and robustness of our algorithm and use an inner loop nested within the optimization minimization iteration to solve the subproblem. Experiments show that under different conditions for blur and salt-and-pepper noise, the proposed method leads to excellent performance in terms of objective evaluation and subjective visual results.