Can anyone here explain to me how New Mexico does their withholding tax? I called them and they weren't very helpful. I just need more information on withholding tax payment frequency.

If you Google "New Mexico tax withholding " you will find all the info, frequency and forms needed. There were 248,000 hits.

I have tried doing that but none of those links explain the different tax liability thresholds and submission dates.

That's because it varies by size of company and payroll, and you can't get it all online.
Why did you say calling isn't an option?