In this paper, we obtain sufficient conditions in terms of projective criteria under which the partial sums of a stationary process with values in $\h$ (a real and separable Hilbert space) admits an approximation, in $\LL^p (\h) $, $p>1$, by a martingale with stationary differences and we then estimate the error of approximation in $\LL^p (\h) $. The results are exploited to further investigate the behavior of the partial sums. In particular we obtain new projective conditions concerning the Marcinkiewicz-Zygmund theorem, the moderate deviations principle and the rates in the central limit theorem in terms of Wasserstein distances. The conditions are well suited for a large variety of examples including linear processes or various kinds of weak dependent or mixing processes. In addition, our approach suits well to investigate the quenched central limit theorem and its invariance principle via martingale approximation, and allows us to show that they hold under the so-called Maxwell-Woodroofe condition that is known to be optimal.