In theory, any private Trello board will connect with Power BI by using the Trello API. In practice this can get quite technical and, frankly, off-putting.
One way to get around this is to work with an offline version of the Trello Board.
When in Trello click on Show Menu:
In the menu click on More…
Then click on Print and Export., then Export JSON
Export the file.
Within Power BI press Get Data but instead of choosing Web, choose JSON. Upload the file you just exported.
From this point on the file will behave exactly as the web connected version with the downside that it is not 100% live. To refresh the data, you would need to export back over the top of the original file.
(image original: http://cdn2.hubspot.net/hubfs/2249672/Imported_Blog_Media/self_learning_feature.jpg)