I encountered some Alikr warriors on the road looking for a redguard woman, which I know is related to the quest "in my time of need". However, upon researching on teh interwebz, it said that the quest is only activated AFTER you do the "dragon rising" quest in the main questline.

The thing is, I haven't even started the main quest yet on this character.

So I was wondering if this quest will become active when I talk to Saadia in whiterun? Or is it just a bug where nothing will happen since I haven't fulfilled the requirements to start the quest?

Saadia can't give you the quest. It's not a bug because it's completely logical: before you talk to the Alik'r, you have no idea about her secret identity, so why would she reveal it out of the blue?

You'll have to start the main quest.

You can get that random encounter before, or after 'Dragon Rising'.

The random encounters are just... random encounters; you'll still have to have "Dragon rising " active in your journal. But, you can get the quest before completing it.

Next time you get the quest, get out of Dragonsreach following Irileth, then enter a shop, get out of the shop and enter another one. On getting out of the second shop you'll find the two Alik'r arguing with the guard while Irileth will be talking to the other guards.

Saadia is so fine...
I'd like to move her into my Orc Stronghold (Adal Matar).

Ah I see....I don't plan on doing the MQ on this character. Looks like they'll just have to keep on searching

I agree that it does seem like an utterly silly quest design. I've just slain a Dragon the size of the Palace and saved the city, and have returned to revel in my triumph, and some guy is bothering me about doing his job for him!?

My guess is that it's there to introduce us to the Thalmor situation. It's the same reason why Whiterun is also the place where we're tasked with rescuing a guy imprisoned by the Thalmor. Bethesda made sure that we don't miss these quests.