Using ADB without MTP support I am using Ubuntu 13.04 with a Motorola Droid Turbo. When I plug the phone in with a USB cable, my phone does show any USB connection notification.

I see that libmtp does not support Motorola's Droid Turbo yet (Ubuntu 13.04 uses libmtp): despite the lack of PTP/MTP support, can I still use ADB?
You should be able to use ADB as far as it is just a tunnel to the Android device's shell.
MTP (Media Transfer Protocol) is not connected to ADB apart from the fact that they share the USB connection.
Just try it out, it shouldn't harm your device.