The fitbit app does not support the Moto X right now. If you check out the fitbit website, even though the Moto X has bluetooth 4.0, Motorola does not have the correct bluetooth libraries installed to communicate with the fitbit app.
Fitbit only supports Galaxy phones. I suggest writing them or commenting in their forums. It appears there are several Android phones that do support BLE but don't work with Fitbit. My guess is that they don't have the development manpower to get things working properly. I think it's important that they realize there is a demand for greater Android support.