My civic makes a quacking noise while im applying the brakes and vehicle weight shifts. A quack that sounds just a duck. What could this be?

 

A duck. let it out of the trunk....

It could be bushings, struts, brakes...a duck...but really, you need to do some inspection to get an idea of what might be worn/failing/loose.

Try pushing down on the rear end while the car is stationary, if it makes the same noise, try to locate the direct source while someone pushes for you.

 

Given that it's duck hunting season, stop driving immediately.

 

This post made me laugh because my civic makes the exact same noise.
Probably a worn bushing or maybe your struts.
Definitely get someone to push the back end up and down while you inspect for the source of the noise.