Unfortunately I think you need to call a professional. BFK will etch quartz, and that definitely sounds like what happened here. They can use quartz renew but it may not get it 100% back.

Oh no! Is there a way to remove this stain? It’s from a carat iron skillet.

Get a professional. 
I have removed rust on granite from cast iron using vinegar and baking soda, but that will also etch quartz.

You're renting. Don't call anyone to fix it without landlord approval. Does your rental agreement allow you to repair it yourself or with a professional? 

 Tell the landlord what happened and they'll figure out a fix that they want and they might bill you for it. 

If you get a professional in there and it worsens then you'll be even more liable for a more expensive fix or a whole new countertop. 

Unfortunately landlords are quite dumb about their property failing to inform renters how they can be proactive and partners in maintaining the home.