Martinez spokesman Mike Lonergan said the governor -- a former Marco Rubio supporter who has not yet said she'll back the GOP nominee in November -- won't be "bullied" into supporting Trump "until she is convinced" he'd act in New Mexico's best interests.
It's the latest escalation between Trump and Martinez, the Republican Party's best-known Latina elected official -- who had said she was too busy to attend Trump's event in her state.
Trump jabbed at Martinez in front of her home-state crowd, blaming her for a growing number of food stamp recipients and for New Mexico's response to the federal government's resettlement of Syrian refugees in the state.
"We have to get your governor to get going. She's got to do a better job, OK? Your governor has got to do a better job," Trump told his crowd in Albuquerque. "She's not doing the job."
He even mused about moving to the state himself to replace Martinez.
"Hey! Maybe I'll run for governor of New Mexico. I'll get this place going," he said. "She's not doing the job. We've got to get her going. Come on -- let's go, governor."
Later, Trump said, "Syrian refugees are being relocated in large numbers to New Mexico. If I was governor, that wouldn't be happening."
It's not just a fight between Trump and a prominent Republican Latina. Martinez is also the chairwoman of the Republican Governors Association, tasked with electing GOP governors this fall, with Trump leading the party on the ballot.
Lonergan suggested Trump doesn't know New Mexico's history or Martinez's work.
"Apparently, Donald Trump doesn't realize Governor Martinez wasn't elected in 2000, that she has fought for welfare reform, and has strongly opposed the President's Syrian refugee plan," he said Wednesday.
Then, he upped the stakes, making clear that Martinez still doesn't back Trump.
"The pot shots weren't about policy, they were about politics. And the governor will not be bullied into supporting a candidate until she is convinced that candidate will fight for New Mexicans," Lonegran said.
"Governor Martinez doesn't care about what Donald Trump says about her -- she cares about what he says he will do to help New Mexicans," he said. "She's disappointed that she didn't hear anything about that last night."
Trump's event in New Mexico was most notable for the protests that erupted afterward, with anti-Trump activists breaking through police barricades, throwing rocks and bottles, setting signs ablaze and in one case smashing a glass door into Albuquerque's convention center, where Trump held his rally.