Alishan's Pendant fully upgrade same as Chaos Horntail?

No. You can't use the drop from HT on the pendant, IIRC a pre-quest is(was*? I haven't cared to scroll HTPs in a while) also needed to exchange for the stone. I remember not being able to even get the quest on alts I didn't want to gollux for.
One is needed for the other, but they're not the same thing.

I remember wasting an egg because I went to Nine Spirit's nest without having the quest active. I stepped on the platform, lost the egg, Nine Spirit didn't have a quest that rewarded the stone to me. It was pretty sad, but that was a long time ago.

But it has changed now, I remember just going up there with the egg and getting a stone pretty recently.