The purpose of this research is to investigate the four dimensions of college L2 learners' engagement with generative AI (GenAI) in translation and the interactive between the four dimensions, in order to provide theoretical and practical references for the optimal application of GenAI in translation education. In the context of the rapid digital transformation in translation education, the engagement with GenAI has become increasingly important for improving teaching the effectiveness. GenAI can effectively enhance cognitive, behavioral, emotional, and agentic engagement in translation by decreasing cognitive load, facilitating proactive operation, evoking positive emotions, and supporting self-regulation, while unregulated GenAI use cause overdependence, academic integrity risks and negative impacts. Current research on GenAI engagement is insufficient in exploring the interaction between the four dimensions of engagement and insufficient in exploring agentic engagement in the context of GenAI-supported translation learning. It also reveals obvious research gaps in cross-dimensional interaction and agentic engagement exploration. This study provides practical guidance on the reasonable application of GenAI in translation teaching, guiding educators, and learners to make effective use of GenAI tools effectively and avoid overdependence and academic integrity risks.