Unfortunately yes if the answers are material to the matter at hand. Here, you were involved in a contempt hearing for alleged failure to pay spousal support. It sounds like the lawyer wanted to us your income and the income of your current spouse to show that there is no reason for you not to pay.
You do not have the right to "plead the 5th" in civil trials unless the answer is reasonably foreseeable to cause you problems in a future criminal trial...i.e. you refuse to answer because your income comes completely from illegal activities.
I agree with Attorney Love. This is a civil trial, not a criminal trial and you can be compelled to testify under most circumstances. Even if it is a criminal act that is the basis for the cause of action. If you refuse to answer, you can be held in contempt. Additionally, you can be asked about your spouses income and may be made to use that money to help pay your obligation. The idea here is to prevent your ex-wife from becoming a "ward of the State".
When you are unsure, you should always speak to an attorney. If you cannot afford one, contact the NC State Bar Association and see if you can find a Pro Bono attorney.
The information provided herein is general information and not designed to create a lawyer-client relationship, nor is it intended to be all-inclusive. You should seek counsel from an attorney about your particular situation to ensure your rights are protected.
Your spouse's income is relevant to the proceedings. I would opt for the on stand as the cross exam would have less prior statements to impeach you with. You really need to find an attorney and see if you can get the amount adjusted.