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The English grammatical error correction system is suitable for the English learning environment, with the goal of accurately correcting errors in learners' writing. However, false corrections are often generated in practical applications, and many errors cannot be corrected, thus misleading learners. The quality estimation model is beneficial to ensure that learners obtain accurate grammatical error correction results and avoid misleading sentences caused by error corrections. Grammatical error correction models can generate multiple hypotheses of higher quality, but existing quality estimation models do not consider interactions between different hypotheses. Based on this, we propose a model based on multiple hypotheses interaction and self-attention, BGANet, for English grammatical error correction quality estimation. BGANet builds interactions between multiple hypotheses, extracts and aggregates grammatical error correction evidence in hypotheses through two kinds of self-attention mechanisms, and evaluates the quality of the generated hypotheses. Experiments on four grammatical error correction datasets show that BGANet has better quality estimation performance.