Okay, so I talked to Vilkas after finishing Glory of the Dead looking for work. I got the "Family Heirloom" look for a long bow in Darkwater Pass.
I go through the cave, kill the "boss" Falmer and get his shaman key. Use it on the chest and there is a long bow. I grab it. The quest does not update. I tried multiple things going back in the cave, 24 hours etc. Finally, I went back to Vilkas.
This is where the glitch is. In my completed quests it shows Family Heirloom (still not updated).
Vilkas mentions the personal Purity quest.
However, I heard that if you accept that it's a glitch and you need to ask Farkas first.
So, I go to Farkas, but he says "Don't you have a job with Vilkas?"
I go to Vilkas looking for work and he also says "You already have a job."
But the said "job" is the not updated Family Heirloom that is in my completed quests section.
What do I do
For PS3, by the way.

It seems you have no choice to do Vilkas purity quest. Skjor usually gives this quest but if he is KIA then Vilkas will take the role. But don't save in case it DOES glitch farkas forever. Sorry I wish I could help more, but the Vilkas/Farkas purity quest bug is quite common and you are right to try to avoid it.

Family Heirloom is just a radiant quest given by Vilkas or Skjorn. The location and enemy are random. So it could be a glitch.

But I thought that Purity of Revenge was supposed to be done before Glory of the Dead. So maybe doing Purity might help you.

If not, reloading an older save would be your only option.

I somehow managed to save myself.
I decided to do the Purity test since it somehow was available despite my current "job".
When I went to the ritual site, Vilkas wasn't there, yet I was able to defeat his wolf form.
I had to go back to the Companion Guild to talk to him, but I was able to successfully complete the quest.
In doing so, this glitch somehow crossed out the other glitch and I can now talk to Farkas.
Whew!
Thanks for your replies!