State tax filing

It's complicated, but not impossible.  There are two distinct steps that must be taken.  First, you must determine the amount of income that is taxable to New York through New York allocation methods.  This will allow the correct amount of NY tax to be calculated and reconciled against the NY withholdings.  The second step will be to claim a credit on your NJ return for taxes paid to NY on the New York allocated income.

New York Return

First, start with declaring the New York income through allocation.  On the W2 entry screen in TurboTax, treat the "NY" and "NJ" Forms W2 as if they are one, because they are.  The second page is additional breakdown of the income reported to New York, but it is part of the same W2.  On the W2 screen, for lines 15-17, enter the NJ information as it appears on your W2,  Your income and the withheld taxes will be transferred to each state's return.

The New York return should be prepared first.  Shortly into the return, you will see a screen entitled New York Allocation.  You will click on No, that not all of your wages were earned in New York State.  The next screen says Your Form W-2 Summary.  Click the Edit link next to your W2 that lists all Federal income as NY income (this is a NY reporting requirement for W-2 wages).  Because the income that must be allocated are vested stock options, New York considers the portion of the stock option obtained during New York-eligible periods during which period the stock options vested.  This is calculated by determining the percentage of work that was taxable to New York between the date on which the options were granted and the date on which the options vested.  The official form used by New York for this Allocation Method (called a Multi-Year Allocation) is Form IT-203-F.

To generate this form, proceed as follows (hit Continue after each step):

1. On the screen Allocate Wages to New York, select the Allocation Method Allocate by Percentage
2. For the next screen, Allocate Wages by Percent, you will enter the percentage of New York income determined by NY Form IT-203-F (click on link for access to the form, which you will fill out manually.  Use Schedule B on page 2 of the form to determine how much of the vested RSU over the vesting period must be taxed by New York.  Then divide that final amount by the NY income reported on the W2 for a percentage, and enter the percentage as 10.00  for 10%, for example.  This step must be done this way to force TurboTax to calculate out the correct income to be taxed.  Save the filled-out form in PDF format)
3. On the screen Your Form W-2 Summary (again) and select Done
4. The next screen asks Did you use Form 203-F to allocate your wages between multiple years?  Select Yes
5. Answer no on the next two screens, and complete the remaining NY interview.
6. Towards the very end of the NY interview, you will be told Looks like you need to attach a PDF.  Click on Edit
7. On the next screen, Upload your PDF, drag and drop the PDF file, or click the Browse Button to find and attach it.  Then click Continue, and Continue again on the repeat of Looks like you need to attach a PDF.

With the proper allocation amounts, the tax withheld should cover the necessary tax (or at least be very close).  Unfortunately, TurboTax does not support calculating IT-203-F, you must do so yourself.

New Jersey Return

You then must make some adjustments on the New Jersey return.  (Note:  New Jersey includes income as taxable that is excluded for Federal and New York income.  Examples include 403(b) retirement deferrals and Section 125 Cafeteria plan deductions).   Eventually you will get to a screen entitled About Your W-2 State Wage Information.  Hit Continue on this screen.  On the following screen, you'll see State ID State Wage Amount (box 16) Remove These Wages.  Click on the NY box to remove those wages.  This is because NJ programming requires software to include all income and allow user to exclude the double-reported income.  It does not allow the software to assume that the income should be excluded.  

Eventually you reach a screen entitled Summary of Taxes Paid to Other States.  If you have filled out the New York return correctly, you should see a credit already calculated here.  If so, you can click Done and complete the New Jersey return as appropriate.  If not, then you can click on Edit, and then enter the amount of income taxed by New York in the box Double-taxed income, and then the amount of net tax you pay to New York on the income (withholdings minus refund reported or in addition to additional tax you must pay on the income to New York).  New Jersey will give you a credit on the lesser of what New York taxes the income or what New Jersey taxes the same income.

This is involved, but it's how to enter this into TurboTax.