Does it pop up automatically each week or do you have to visit a quest giver?
It’s available each week from an NPC called Chronicler Toopa and Shoopa (Alliance / Horde)
This might help more
But some weeks its not available?
Sorry, I have no idea why it would only appear every 2 weeks. Are you sure you’re not just handing it in after the reset?
This week I didn’t get a bg quest but a 4 mythics quest instead.
I don’t think the quest is for BGs specifically, just a weekly to do x4 of something 
Weekly quests differ, you can see what will be the weekly ‘event’ on the in game calendar.
This week is the BfA mythic dungeon week; there are also battleground, timewalking, world quest, pet battle weeks.
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