Automated machine learning (AutoML) is a promising solution for building a machine learning framework without human assistance and has attracted significant attention throughout the computational intelligence research community. Although there has been an emerging interest in graph neural architecture search, current research focuses on the specific design of semi-supervised or supervised graph neural networks. Motivated by this, we propose a novel method that enables the automatic construction of flexible self-supervised graph representation learning frameworks for the first time as far as we know, referred to as AutoSGRL. Based on existing self-supervised graph contrastive learning methods, AutoSGRL establishes a framework search space for self-supervised graph representation learning, which encompasses data augmentation strategies and proxy tasks for constructing graph contrastive learning frameworks, and the hyperparameters required for model training. Then, we implement an automatic search engine based on genetic algorithms, which constructs multiple self-supervised graph representation learning frameworks as the initial population. By simulating the process of biological evolution including selection, crossover, and mutation, the search engine iteratively evolves the population to identify high-performed frameworks and optimal hyperparameters. Empirical studies demonstrate that our AutoSGRL achieves comparative or even better performance than state-of-the-art manual-designed self-supervised graph representation learning methods and semi-supervised graph neural architecture search methods.