All is guessing, but if the expansion adds new content without eliminating previous one, the relic quest will probably exist, only made quicker on the last steps that are the only ones that are not relatively quick to do. The relic weapon, at least the first part, is nearly part of the main story since the job quests speak about it, it drives you to do the HM primals and some dungeons too.

I suppose the ones who have the last step of the relic will have some advantage for the new weapon quest in Heavensward, if it is not required for it. The relic will probably be quite quick to do, but it's playable content, why eliminate it?

Up to now there has been always a grindy but not very difficult way to get a good weapon (relic), and they also have added other ways to get good weapons (the ilvl110 ones are fine for all content except maybe the hardest now), and the best one is the drop of the hardest raid. Usually people who do that hardest raid when it is new content have the latest upgrade of the relic or are working on it.

Probably there will be a similar quest, it's like a tradition, and there will be other ways to get good weapons apart from the also traditional drop of the last hardest instance. As the game advances, there are more and more options.