N.M. Stat. § 39-3-28
The supreme court or court of appeals, on the determination of a cause on appeal or error, may award execution to carry it into effect, or may remit the record with its decision to the district court from which the cause came, and the determination shall be carried into effect by the district court. When any writ of execution sued out of the supreme court or court of appeals is placed in the hands of any officer for levy or collection and the officer fails to find any property from which it may be satisfied, the officer shall notify all persons who may be indebted to the defendant named in the writ not to pay the defendant, but to appear before the district court from which the cause was originally taken by appeal or writ of error and answer on oath concerning his indebtedness. Thereupon, like proceedings shall be had in the district court as in case of garnishees summoned in suits originating by attachment in the district courts.
NMS § 39-3-28