In this letter, we analyze the error rate performance of M-ary coherent free-space optical (FSO) communications under strong atmospheric turbulence. Specifically, we derive the exact error rates for M-ary phase shift keying (MPSK) and M-ary quadrature amplitude modulation (MQAM) based on moment-generating function (MGF) with negative exponential distributed turbulence, where maximum ratio combining (MRC) receiver is adopted to mitigate the turbulence effects. Additionally, by evaluating the asymptotic error rate in high signal-to-noise ratio (SNR) regime, it is possible to effectively investigate and predict the error rate performance for various system configurations. The accuracy and the effectiveness of our theoretical analyses are verified via numerical results.