Optimal-REQUEST is a recursive algorithm for least-squares fitting of the attitude quaternion of a rigid body to vector measurements. It relies on the knowledge of the variances in the measurement and process noises and is therefore prone to divergence due to modeling errors. The algorithm presented here is an adaptive Optimal-REQUEST procedure, based on the idea of covariance matching, which adjusts the noise variances in the filter in an on-line and optimal manner. For this purpose, non-classical residuals are designed by exploiting the structure of the so-called K-matrix and their statistical properties are investigated. As a result, although processing the same vector observation, two distinct algorithms can be developed for measurement noise adaptive filtering and for process noise adaptive filtering. The special case of zero-mean white measurement and process noises is considered. A simulation study is used to demonstrate the performance of the various adaptive algorithms. Extensive Monte-Carlo simulations show that the process noise adaptive procedure can compensate for large unknown biases in the process noise.