You must mean the quill, that's a tough one to answer without doing a demo.
If you are wrapping the feather 'Palmer' style then you are tying it in at the rear by the tip and winding forward. That quill will become thicker as you get closer to the base of the feather. You either need to find hackles with really long fibers (like Schlappen) or find a way to avoid wrapping all the way to that thick butt end of the quill.
Try to make the point where you tie the feather off on the bottom and use plenty of thread tension to bind it down as tightly to the hook and body materials as you can.
The scissors are part of the result, what type do you have? I have been using the same pair of Dr. Slick curved blades for over 20 years and they'll still trim a quill pretty close. After you have the quill strapped down tight, be careful not to cut your thread when trying to get close. Oh, you already did that huh
If not just remember it is a danger and is always a bummer when it happens.
So: Good long hackle fibers; Plenty of tension at the tie down; Good scissors and be careful; After you have it trimmed put a few more really tight wraps over the butt.
The only other trick I could point you to is to soak the hackles. This will soften the quills and they won't want to splinter or break on you (hardly ever) Learning to tie with wet hackles is another part of fly tying but I do it every time I tie. Every salmon fly I make I use wet hackle so that I don't have to deal with either a feather breaking when I'm almost done, or those real stiff butts that just don't want to tie down.
Keep me posted on whether any of this turns out to be helpful.