If I lived part-year in NJ but worked in NY and no NJ income tax was deducted (I only paid NY taxes), do I owe NJ money?

In 2021, I lived in NJ for 8 months and NY for 4. I kept the same NY-based employer the whole year. No NJ-state tax was deducted, but now it says I owe NJ money, unless I claim the tax credit for "double-taxed income." Can I claim the tax credit? I already paid NY state tax.