I studied in NYC at university and sublet an apartment while I was working at an internship in the city. My permanent residence is in Illinois.
Does this mean that I "maintained living quarters" in NYC?
No, maintaining living quarters means you keep the abode even when you are not physically in the city- so if you paid the rent all year but did not live in the apartment or sublease, you maintained living quarters.
In this case would she have to pay NY state tax?
If I temporarily subletted (rented) several rooms in NY while working remotely would that mean I would owe NY tax if I was a resident of another state?
If so, do I need to put the exact address of the multiple sublets or just that it was in NY for my tax return? The landlords did not give official approval for the tenant to sublease so not sure if that would cause any issues.
Yes, as a resident of another state and non-resident for NY you would owe NY state tax on income earned for work carried on in NY. Here is the NY reference.
The address on your tax return is the address where you receive your mail on a regular basis. Typically that will be in your permanent resident state. The address on you tax return is not shared with anyone; it is for tax correspondence purposes not residency determination.
To generate the proper returns in TurboTax in the "MY Info" indicate that you lived in your permanent resident state on 31 Dec and if you did not have another resident state in the tax year indicate "No" you did not live in another state. Still in the "MY Info" section scroll down to "Other State Income" and answer "Yes" you earned income in another state and select NY from the dropdown list. This will generate any resident returns needed as well as a NY non-resident return.