Research on GenAI-assisted language learning has grown, yet learner engagement in GenAI-assisted interpreting training remains underexplored. This exploratory study examined how nine participants engaged with GenAI across behavioural, cognitive, and emotional dimensions in an 8-week training program. Data were collected through an engagement scale, video recordings, open-ended questionnaires, and reflective diaries. Results show that GenAI fostered emotional engagement, particularly enhancing confidence and self-efficacy. At the cognitive level, it reduced note-taking load, allowing participants to focus on evaluating AI-generated output. Behaviourally, participants invested effort in post-editing to improve language quality. Given the small, all-female sample, findings provide preliminary evidence rather than generalisable conclusions. The study suggests potential for integrating GenAI into interpreter training while highlighting the need for further large-scale research.