The trails around Gomez Peak are shared use and more like backcountry hiking trails than mountain bike trails, but they make for a good technical mountain bike ride with some hard climbing. The climb to the top is well worth the views and the mountain bike ride back down.
Boston Hill Mountain Bike Trails are located in Silver City New Mexico in what seems to be old mining land that was reclaimed for public use. It’s a nice use of space and had some nice trails to ride.
Cedar Creek Mountain Bike Trails are actually a shared used trail system with hikers. However, there is one dedicated downhill mountain bike flow trail within the Cedar Creek Trail System in Ruidoso, New Mexico.