If someone was convicted of a felony 20 years ago. Can they ever own a firearm in NEW MEXICO?

It is unlawful, without prior approval from the bus company, to board or attempt to board a bus while in possession of a firearm upon the person or effects and readily accessible while on the bus. This does not apply to duly elected or appointed law enforcement officers or commercial security personnel in the lawful discharge of their duties.

It is unlawful for a felon to receive, transport, or possess a firearm.

Felon is defined as:

A person who has been convicted of a felony offense by a court of the United States or of any of its states or political subdivisions;
And less than ten years have passed since the person completed serving his sentence or period of probation for the felony conviction, whichever is later;
And the person has not been pardoned for the conviction by the appropriate authority;
And the person has not received a deferred sentence.



ETA:
Believe crime was convicted in East Coast.

depends.  state or federal?

Yes if pardoned or had their rights restored in the state that holds the conviction.

Edit after the OP edit - it appears that rights are "automatically" restored in NM after 10 years.  It appears that it would be wise to verify this with a competent attorney in NM familiar with this area of the law.

person lives in New Mexico. Believe crime was robbery or burglary.

if it's state felony, and assuming no pardon has taken place, he would have to get it expunged if that crime was able to be expunged.  if it's federal, the only option is a pardon.

State: depends on state



Federal: Theoretically yes, but Congress has defunded the branch of the ATF that does this.  So really no.






 

I've heard in rare cases you can appeal for your firearms rights back but unlike getting your voting rights back I'm not sure how often this happens

Yes. If your rights are restored under state law, then they are also restored under Federal law. The state where the conviction occurred is the state whose law controls.