DIAGNOSIS That's almost certainly coil whine coming from the power adapter (the "brick" on the plug). It's a high-frequency noise caused by electrical components vibrating, and it's harmless but annoying.
FIX The fix is to replace the faulty power adapter. 1) Unplug the buzzing adapter from the wall and from the camera's weatherproof cable. 2) Contact Google Nest support directly. They are aware of this issue with some batches of outdoor power adapters and will often send you a replacement for free if your camera is under warranty. Have your camera's serial number ready from the Google Home app.
IF THAT DIDN'T WORK If a new adapter from Google still buzzes, the issue could be "dirty" power from your outdoor outlet. Try plugging the camera into a different outdoor circuit, or use a simple, high-quality surge protector (not a power strip with filtering) between the outlet and the adapter.
HEADS UP Don't try to open or fix the buzzing adapter yourself. It's sealed for weatherproofing and contains capacitors that can hold a dangerous charge even when unplugged. Just get it replaced.