I know it is uncommon but the company that hired me pays the state tax on their income from contracts.
What do you mean "uncommon"? Any business that has a physical presence in NM is required to register with the state and pay tax on their business income.
So am I exempt from state income or gross receipts tax?
Why are you asking this question? I'm not comprehending what could make you think you don't have to pay state taxes, just like every other business in NM does.
I suspect you're confusing apples and peanut butter here, but I could be wrong. Generally the "gross receipts" tax is the sales tax on products & services sold. This is totally and completely separate from both federal income taxes and state income taxes.
Generally for both state and federal taxes, the taxable business income is your gross business income minus deductible business expenses.