The problem of adaptive filtering for the two-state telegraph process observed in white Gaussian noise is considered. We assume that the infinitesimal matrix depends on two unknown parameters. The construction of the adaptive Wonham filter begins with the estimation of these parameters based on observations over a relatively small learning interval using the method of moments. These estimators are then used to define the One-step MLE-process. Substituting this estimator process into the filtering equation provides the adaptive filter for the conditional expectation of the unobserved component. We then consider a two-state process based on the adaptive filter as an estimator for the telegraph process and describe the asymptotic behavior of the estimation error. The asymptotic properties of the MME, MLE, and BE estimators are also described.