Page-oriented volume holographic memories (POVHM) is a quadratically nonlinear channel because of the intensity detection at its output. To combat the two-dimensional intersymbol interference in a high-capacity POVHM, equalization of the output intensity array requires identification of the quadratic and possibly spatially varying impulse response of the channel. Although conventional adaptive filtering schemes are devised for identification of linear channels, they also require the length of the impulse response to be known in advance. In this work, we develop multistage quadratic normalized least mean square (LMS) (MS-QNLMS) adaptive filtering and multistage Volterra normalized LMS (MS-VNLMS) filtering to estimate the channel under quadratic nonlinearity, which do not require the support or length of the impulse response to be known a priori. By employing extensive numerical experiments, we provide performance and convergence comparisons of the proposed schemes with respect to a true-order quadratic estimator. We also show that MS-QNLMS filtering has less computational complexity and converges faster and more robust to various channel parameters as compared to MS-VNLMS.