It may or may not. Early models of the Bose A20 headset do not support music via Bluetooth, even though the headset will allow cellphone connections. Since the Runway Proximity Advisor™ alerts play via music, they can't be heard on the early Bose A20 headsets via Bluetooth.
Starting in 2015, Bose updated the A20 headset to support music via Bluetooth, allowing ForeFlight alerts to be transmitted via Bluetooth to the headset. With this update, if you are playing music via Bluetooth and ForeFlight Mobile issues an audible alert, the music volume lowers until the alert is finished. Then the music volume ramps up again.
If you have an early model (pre-2015 update) Bose A20 headset, you can have it updated to support music via Bluetooth. Bose sells a version 2 cord assembly that can be shipped to you to replace your version 1 cord assembly. Please contact Bose at 800-233-4416.
Alternatively, you can purchase a male-male 3.5mm cord and plug the iPad's headphone jack into the 3.5mm port on the A20 control box, or if your audio panel has an audio-in port, you could plug it in there.
Finally, some pilots have reported that using the Blulink Adaptor works well to add Bluetooth music to their headset.