I believe there's a general statute against harassing wildlife in NM, but I couldn't find a good reference.
There are pretty strict requirements for using dogs to chase animals in NM as listed in the
rule book here. While it doesn't specifically state that you can't let your dog chase wildlife, it provides enough specific conditions that one could infer chasing (an activity not specified) is not allowed.
It would definitely be illegal on NM state land:
Originally Posted By: nmstatelands.orgProhibited Activities:
1) disturbing, harassing, injuring, destroying or removing wildlife, livestock, or any other animal life;
It appears to be illegal on BLM land as well based on this:
Originally Posted By: Watchable Wildlife page on BLM's NM page•Never chase or harass wildlife; harassment of wildlife is against the law