You sure can! This is different across handset models, but the "notification volume" is what you are looking for. When you adjust the volume on your handset, you should see a button to expand the slider to show all your volume options, such as notifications, call volume and media.
We find that 50% is a good level for ringtones/notifications on most handsets, while media is usually best at 70-90%. Unfortunately it varies a lot across handset makes and models; even an Android version update can change your handset's bluetooth audio output.
Keep in mind that your BT controller will change whatever is currently making sound. So if it's ringing, the controller will change your ringtone volume. If you're on a call, it'll change your call volume. If you're listening to music, or hearing a navigation, it'll change media volume.
When you're not receiving any audio output, unfortunately that's handset specific again - some will change ringtone volume, others will do nothing.