How do I "think in QUnit" with my own JavaScript libraries?

Travis-ci is indeed a good way to trigger tests, including for client-side libraries.

The answer "Using Travis-CI for client-side JavaScript libraries?" give a good examples of such tests, in kort/kort/tree/develop/test/client, which includes QUnit in its index.hml file.
The key is to use Grunt.js.

An example of Travis job: kort/kort/jobs/3266208 does include running QUnit.

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4 years later, update following the bounty:

You can see unit tests maintained overtime in the FredrikNoren/ungit, which uses Travis.

But since you are not on GitHub, you can use alternatives like GitLab-CI (using you very own GitLab CE -- Community Edition -- server if you want)

See more with that approach in "GitLab Frontend testing standards and style guidelines": those best practices applied to the GitLab project itself could by applied to your projects as well.
You can see their unit test policy.