The Bose-Einstein condensate (BEC) provides a controllable platform for studying nonlinear quantum dynamics, including Josephson oscillations, macroscopic quantum self-trapping (MQST), and quantum coherence. In a two-component BEC with Rabi coupling, quantum fluctuations become important when the mean-field interactions are strongly suppressed. Here, we investigate the effect of the Lee-Huang-Yang (LHY) correction on the Rabi dynamics of a Rabi-coupled two-component BEC. Starting from the three-dimensional Gross-Pitaevskii equations including Rabi coupling and the LHY correction, we consider the case g_12=-\sqrtg_11g_22, for which the leading mean-field interaction is suppressed. Within the single-mode approximation, we derive an effective two-mode Hamiltonian in terms of the population imbalance z=(N_1-N_2)/N and the relative phase \phi=\theta_2-\theta_1. The fixed points and their linear stability are then analyzed, and the corresponding small-oscillation frequencies are obtained. Numerical simulations are used to verify the analytical results and to illustrate the dynamics in terms of an effective potential. For \gamma=1, the LHY contributions from the two components cancel, and the usual Rabi dynamics is recovered. For asymmetric interactions, taking \gamma=1.5 as an example, the LHY correction changes both the number and stability of the fixed points. For the \phi_s=0 branch and \Lambda_\rm MF>\Lambda_\rm MF^C, two stable fixed points and one unstable fixed point exist when \Lambda_\rm LHY<\Lambda_\rm LHY^C. At \Lambda_\rm LHY=\Lambda_\rm LHY^C, the stable and unstable branches coalesce and the corresponding eigenvalue vanishes. Above this critical value, only the unstable fixed point remains. The LHY correction can therefore induce a transition between MQST and large-amplitude Rabi oscillations. For example, for \gamma=1.5 and \Lambda_\rm MF=0.4, the critical value is \Lambda_\rm LHY^C\approx0.0737. At \Lambda_\rm LHY=0.0707, the population imbalance and relative phase remain bounded, corresponding to MQST. When \Lambda_\rm LHY is increased to 0.0767, the relative phase becomes unbounded while the population imbalance undergoes large-amplitude periodic reversals, indicating Rabi oscillations. This transition results from the partial cancellation between the asymmetric LHY contribution and the mean-field nonlinearity, which reduces the effective-potential barrier and allows the previously trapped motion to escape. On the \phi_s=\pi branch, the LHY correction produces only a weak change in the dynamics, and no qualitative change in the phase-space structure is observed. It also modifies the small-oscillation frequencies: the frequency on the \phi_s=\pi branch increases gradually with \Lambda_\rm LHY, whereas the \phi_s=0 branch develops two frequency branches for \Lambda_\rm MF>0.3, corresponding to the two stable fixed points. The relevant parameter regime is accessible in current ultracold-atom experiments. This study deepens the understanding of nonlinear dynamics in multi-component condensates regulated by quantum fluctuations.