I’m not an ornothologist, so take that under advisement.
I’d assume you could describe it like you would describe anything. A fight is a time of high emotion, high stress, etc, I would focus on those feelings instead of trying to be technical. It’s important that the reader feel those feelings of fear, worry, stress, anger and whatever else your character is feeling.
Second, I’d spend a lot of time watching nature documentaries dealing with your subject and reading books specific to the kinds of birds you’re writing. There are probably a host of YA, Middle Grade, and possibly even adult books that have written stories from the perspective of birds and other animals (though I can’t remember what they are), you might also gain inspiration from reading what others have written. Ask your local librarian to see if he or she can give you a good place to start. Remember, you want technical books as well as fiction.
I know this advice is general, but the best answers are the ones you will come up with for yourself. The best way to accurately represent a fight between to birds is to watch birds. If you don’t have cats, set up a few bird feeders in your backyard that are close to a window that will provide easy viewing. Watch and take notes on how they interact with each other, gestures of affection in family groups, and even the fights that sometimes break out between different species. If you’re lucky, you’ll live in a place where you can attract all manner of birds.
It may take some time, but if you feed them, they will come. By watching, reading, and studying up on birds you will better come to understand their perspectives. When you understand their perspective, you can write them in all manner of different situations.