Yes, you are taxed in both states on this income. Virginia taxes you as a nonresident, and New Jersey as a resident (resident states tax all of your income regardless of where you earn it). However, to eliminate the penalty of double-taxation, New Jersey will give you a credit for the tax you must pay to Virginia on this income. For this reason, you will want to prepare the Virginia return first, to ensure that NJ grants the credit. You will be able to verify this credit on the New Jersey portion of the return which addresses double-taxed income.
This FAQ addresses this subject a bit more in-depth: https://ttlc.intuit.com/replies/3301744
Yes, you are taxed in both states on this income. Virginia taxes you as a nonresident, and New Jersey as a resident (resident states tax all of your income regardless of where you earn it). However, to eliminate the penalty of double-taxation, New Jersey will give you a credit for the tax you must pay to Virginia on this income. For this reason, you will want to prepare the Virginia return first, to ensure that NJ grants the credit. You will be able to verify this credit on the New Jersey portion of the return which addresses double-taxed income.
This FAQ addresses this subject a bit more in-depth: https://ttlc.intuit.com/replies/3301744