No, New Mexico does not have a similar provision. However, some of the Rules in New Mexico can be read together to try to achieve a similar result, although it is more in the court's discretion than required by rule like in Illinois.
Rule 1-026(B)(3) states:
(3) Witnesses and exhibits. Parties may obtain discovery of the identity of each person expected to be called as a witness at trial, the subject matter of the witness’s expected testimony and the substance of the witness’s testimony. Parties may also discover the name, address and telephone number of each individual likely to have discoverable information that another party may use to support its claims or defenses as well as the subjects of such information. Parties may obtain a copy of, or a description by category and location of, all documents, electronically stored information, and tangible things that a party may use to support its claims or defenses.
Rule 1-033 governs interrogatories. The information stated above can be requested in an interrogatory, including the substance of a witness's testimony.
Rule 1-037 governs sanctions. If the party doesn't cooperate with discovery and any court order ordering discovery, then a party can make a motion to exclude that evidence/information from trial. So if a party has properly requested information and it was not provided by the other side, the court has the ability to keep that information out of trial.
So it is not identical to the rule in Illinois, nor as easy, but it is possible to use multiple rules to achieve a similar outcome.