At least here in NJ, a road can be a private road or it can be county road - but it cannot be both.
If you have a private road on your property, then you have the right to exclude others from it.
If there is a county road that bisects your property, then you cannot exclude others from using a public roadway.
I am not a NM attorney, laws vary from state to state, therefore you should always consult a local attorney.
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If this answer was helpful, please mark it as helpful or as a best answer. This answer is for general education purposes only. It neither creates an attorney-client relationship nor provides legal guidance or advice. The answer is based on the limited information provided and the answer might be different had additional information been provided. You should consult an attorney.