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A novel addition to the family of integral transforms, the quadratic phase Fourier transform (QPFT) embodies a variety of signal processing tools, including the Fourier transform (FT), fractional Fourier transform (FRFT), linear canonical transform (LCT), and special affine Fourier transforms. Due to its additional degrees of freedom, QPFT performs better in applications than other time-frequency analysis methods. Recently, quaternion quadratic phase Fourier (QQPFT), an extension of the QPFT in quaternion algebra, has been derived and since received noticeable attention because of its expressiveness and grace in the analysis of multidimensional quaternion-valued signals and visuals. To the best of our knowledge, the discrete form of the QQPFT is undefined, making it impossible to compute the QQPFT using digital techniques at this time. It initiated us to introduce the two-dimensional (2D) discrete quaternion quadratic phase Fourier (DQQPFT) that is analogous to the 2D discrete quaternion Fourier transform (DQFT). Some fundamental properties including Modulation, the reconstruction formula and the Plancherel theorem of the 2D DQQPFT are obtained. Crucially, the fast computation algorithm and convolution theorem of 2D DQQPFT which are essential for engineering applications are also taken into account. Finally, we present an application of the DQQPFT to study the two-dimensional discrete linear time-varying systems.