In this paper, short-term forecast of wind farm generation is investigated by applying spatio-temporal analysis to extensive measurement data collected from a large wind farm where multiple classes of wind turbines are installed. Specifically, using the data of the wind turbines' power outputs recorded across two consecutive years, graph-learning based spatio-temporal analysis is carried out to characterize the statistical distribution and quantify the level crossing rate of the wind farm's aggregate power output. Built on these characterizations, finite-state Markov chains are constructed for each epoch of three hours and for each individual month, which accounts for the diurnal non-stationarity and the seasonality of wind farm generation. Short-term distributional forecasts and a point forecast are then derived by using the Markov chains and ramp trend information. The distributional forecast can be utilized to study stochastic unit commitment and economic dispatch problems via a Markovian approach. The developed Markov-chain-based distributional forecasts are compared with existing approaches based on high-order autoregressive models and Markov chains by uniform quantization, and the devised point forecasts are compared with persistence forecasts and high-order autoregressive model-based point forecasts. Numerical test results demonstrate the improved performance of the Markov chains developed by spatio-temporal analysis over existing approaches.