With the continuous development of large-scale complex hybrid AC-DC grids, the fast adjustability of HVDC systems is required by the grid to provide frequency regulation services. This paper develops a fully data-driven linear quadratic regulator (LQR) for the HVDC to provide temporal frequency support. The main technical challenge is the complexity and the nonlinearity of multi-infeed hybrid AC-DC (MIDC) systems dynamics that make the LQR intractable. Based on Koopman operator (KO) theory, a Koopman eigenpairs construction method is developed to fit a global linear dynamic model of MIDC systems. Once globally linear representation of uncontrolled system dynamics is obtained offline, the control term is constituted by the gradient of the identified eigenfunctions and the control matrix $\mathbf{B}$. In case that $\mathbf{B}$ is unknown, we propose a method to identify it based on the verified Koopman eigenfunctions. The active power reference is optimized online for LCC-HVDC in a moving horizon fashion to provide frequency support, with only local frequency and transmission power measurements. The robustness of the proposed control method against approximation errors of the linear representation in eigenfunction coordinates is analyzed. Simulation results show the effectiveness, robustness and adaptability of the proposed emergency control strategy.