In the Retail client, this option was toggled in the Minimap.
I cannot find it in the Classic Client.
Yes, you cannot track quests on the minimap in Vanilla, but when you were in range of the NPC which did have them, they would have the exclamation point on their heads. This is the part that is missing.
It’s only because I have memorised the handful of low level quests inside Stormwind that I knew the quests were there - and after checking the Interface options I couldn’t locate the option.
That option didn’t exist in vanilla, though it is one I wish they would put in.
Download Questie, you will see them
This option wasn’t added until TBC I believe.
It’s clearly been a while. 
Cheers for the responses, guys!
Is there an add-on that would replicate this? just making it so I can see the ! above their head is all I’m looking for, I don’t want to have to spend hours trying to turn everything else that Questie does off just for that one lil feature.
If you don’t like questie, or you don’t want to use an addon, you can view it as a way to encourage exploration. Its one of the things I loved about Classic. It felt like an exciting little treat when I found a quest hidden away some where.
So your advice is to talk to every single NPC I see to check if they offer a quest that I can’t see?
Why would you have to do that? You can see the ! over their heads. And its not advice for you, or anyone in particular, just another option.
Edit: I misunderstood and was responding to the OP. I thought he was looking for the ! on the minimap. No, there is no way to turn on the ! over someone’s head for low level quests, addons is all you have for that.
When you are higher level than is intended for the quest (10 levels above I think) the ! is not shown above their head anymore. I was hoping to find an add-on that made it so they still showed up, and only that.
Yeah I edited my post once I realized what you were talking about. It doesn’t change what I was saying before though, you can use addons or you can run around and talk to everyone. Those are your choices.
So does anyone know of an add-on that allows you to see the ! above the head of a quest giver for low level quests?
Trying to figure out how to make an add-on for this is going extremely poorly.