As Sheldon says, short answer is no. If the agency supplies a GTFS-realtime Service Alerts feed, that can be fed into the local OneBusAway server for that region. But if they don't, then there is no (simple) way to get the existing alerts info into OneBusAway (or Google Maps, or any other third-party app).
So, if the agency you're referring to is one that doesn't supply alerts in the GTFS-realtime format, I'd strongly encourage you to voice your preference to the agency that they start!
Sean