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(Three years later)
There is a way to do this now, but it's still not perfect.
See the "PR triggers" section of this document: https://docs.microsoft.com/en-us/azure/devops/pipelines/repos/bitbucket?view=azure-devops&tabs=yaml#pr-triggers
Unfortunately, the PR trigger only works when a PR is created or updated, but not on the merge commit. The Azure documention reference above says:
Each new run builds the latest commit from the source branch of the pull request. This is different from how Azure Pipelines builds pull requests in other repositories (e.g., Azure Repos or GitHub), where it builds the merge commit. Unfortunately, Bitbucket does not expose information about the merge commit, which contains the merged code between the source and target branches of the pull request.
I'm searching the Atlassian Community right now for more information on that last sentence. (That's how I ran across this question.)
Currently there doesn't seem to be an existing plugin between Azure Pipelines and Bitbucket. There is one for Jira as you can see here.
You can however set up Bitbucket Pipelines in your Bitbucket repository to trigger Azure deployments on your PRs. We also have Bitbucket pipes which wrap up this functionality for you. Here are some docs you might find useful: