The windowed quadratic phase Fourier transform (WQPFT) combines the localization capability of windowed transforms with the quadratic phase modulation of the quadratic phase Fourier transform (QPFT). The main contribution of this paper is the development of convolution theorems for the WQPFT in both the spatial and spectral domains, together with a comprehensive theoretical framework for the transform. Two versions of the inversion formula are presented: a double integral form and a univariate integral form. Fundamental properties of the WQPFT are established, including the uniform continuity, reproducing kernel, and the associated localization operator. We further prove existence results and derive norm estimates for the convolution operations associated with the WQPFT. As applications, we discuss multiplicative filter design using the WQPFT and the solvability of a class of convolution equations. In addition, we obtain analogues of the Poisson summation formula, sampling theorem, and Paley-Wiener criterion, and outline potential applications of the WQPFT.