There is no such DBus API at the moment, but it is on the roadmap.
However, it is possible to find out by using the GetJwtToken DBus API to get the JWT token and server, and then use that to call the server manually. The response will contain the next update name.
Note that I think the backend will consider the update as having started for that device if you call that endpoint. But this only affects the information in the UI, it doesn’t have any practical effect, as far as I know.