By using the quantum Yang-Baxterization approach to the three different Hamiltonians, we investigate the behavior of the quantum Fisher information (QFI) under the actions of these Hamiltonians on the different two-qubit input states and by estimating the meaningful parameter phi. We address the overall estimation properties by evaluating the QFI for the whole system undergone different unitary evolution. The results show that the behavior of the QFI depends on the choice of the initial states. Choosing the optimal input states can improve the precision of quantum parameter estimation. On the other hand, we also focus on the dynamical evolution of QFI to distinguish Markovianity and non-Markovianity of the process by adopting the flow of QFI as the quantitative measure for the information flow. We show that the Hamiltonians constructed with Yang-Baxter matrices influence the dynamics of the system in the sense of the Markovianity and non-Markovianity. In certain ranges of parameters, we observe that dynamical evolutions of the systems show non-Markovian behavior in which the information flows from the environment to the system.