To alleviate the impact of atmospheric turbulence on receiver carrier recovery in free-space optical communication, this study introduces a novel frame synchronization and carrier recovery method. This method employs a blend of pseudorandom sequences and specific cyclic quadrature phase shift keying (QPSK) training sequences, facilitating frame synchronization and frequency offset (FO) estimation. The research designs a time-series metric curve to counteract the effect of side peaks, enhancing the accuracy of frequency offset estimation via QPSK spectrum features. Furthermore, the method incorporates a two-stage frequency offset estimation process, utilizing the inherent sign characteristics of the x and y polarization states. Applied to the quadrature amplitude modulation system under atmospheric turbulence conditions, the proposed algorithm demonstrates high precision under both strong and weak turbulence conditions. Furthermore, we confirmed the universality of the algorithm across multiple modulation formats.