In short, I work in New York but Live in New Jersey. According to New
York state law I need to pay tax for the income I owned there,
however, my company withheld my state tax for New Jersey instead of
New York. There is no information about New York on my W2. When I
tried to use TurboTax to file my tax, after entering all the W2
information, it tells me that I don't own tax in New York, which is
probably wrong but I am not 100% sure. The reason TurboTax tells me
this is understandeable because that there is nothing on my W2 about
New York. Does anyone know how to handle this situation? The deadline
of the filing is closing.

Prompt help will be much appreciated.

Fill out the NYS tax forms and determine your NYS tax.  You will then
subtract that tax from your NJ tax due.  You may have to do it
manually.