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Bayesian inference plays a vital role in Structural Health Monitoring (SHM) by assessing structural integrity through probabilistic model updating using monitoring data. A crucial component in Bayesian inference is the evaluation of the likelihood function. For some situations, the likelihood function is not available in closed form or is analytically intractable, due to either a computationally intractable forward model solution or lack of measurements for some inputs. While likelihood-free Bayesian inference methods such as approximate Bayesian computation have been proposed to tackle this issue, the required computational effort is high, and the accuracy relies on similarity criteria, which makes these methods unsuitable for online monitoring. This study investigates a novel likelihood-free and computationally efficient Bayesian inference method, named BayesFlow, for probabilistic damage inference in SHM through model updating. It consists of a training phase and an online monitoring phase. In the training phase, BayesFlow approximates the posterior distributions of structural parameters by jointly training a conditional invertible neutral network (cINN) and a summary network. The cINN connects structural parameters with a latent of the posterior distribution. The summary network automatically learns the maximally informative statistics for model updating from data of output variables rather than hand-crafted features. In the online monitoring phase, BayesFlow directly predicts the posterior distribution for any given observations, without computing any likelihood or evaluating the forward prediction model, and thereby allows for real-time monitoring. Two benchmark examples, including an 18-story steel frame and a concrete building frame, are used to demonstrate the proposed method. Results comparison of the proposed method and the Differential Evolutionary Adaptive Metropolis (DREAM) sampling method demonstrates advantages of the proposed method in terms of both accuracy and computational efficiency.