Can I run a workflow when a contact moves to a pipeline stage and send the data to n8n?

Yes. Use the Pipeline Column Changed trigger. Each trigger is set for one stage, and selecting several stages in a single trigger is not available. A workflow can have multiple triggers, so add one trigger per stage you care about (for example Qualified and Proposal) in the same workflow, and it runs when any of them fires.

To send the data to another system such as n8n, add an API step that posts to that system's webhook. Every step, including the trigger, makes its output available as variables, so you can pass the stage change data along and the receiving tool can filter on which stage the contact moved from and to.