We propose a fast and accurate signal quality monitoring scheme that uses convolutional neural networks for error vector magnitude (EVM) estimation in coherent optical communications. We build a regression model to extract EVM information from complex signal constellation diagrams using a small number of received symbols. For the additive-white-Gaussian-noise-impaired channel, the proposed EVM estimation scheme shows a normalized mean absolute estimation error of 3.7% for quadrature phase-shift keying, 2.2% for 16-Ary quadrature amplitude modulation (16QAM), and 1.1% for 64QAM signals, requiring only 100 symbols per constellation cluster in each observation period. Therefore, it can be used as a low-complexity alternative to conventional bit-error-rate estimation, enabling solutions for intelligent optical performance monitoring.