Mr. Molina has repeatedly maintained his innocence, and his supporters say the accusations against him originated with complaints from political opponents who had backed Mr. Garcia’s re-election.

Mr. Molina was charged with two felony counts of illegal voting and one felony count of engaging in organized election fraud. Ms. Molina, 42, was charged with one count of illegal voting.

“The mayor is innocent of what he’s being accused of,” said Mr. Molina’s lawyer, Carlos A. Garcia. “My client unseated an incumbent in the last election. As a result of that, he went in and made some changes. There were and are several people who are upset at my client, and it’s our position this is all politics. There’s a power struggle in Hidalgo County, specifically in Edinburg.”

Cary Zayas, the spokeswoman for the City of Edinburg, said in a statement defending the mayor that the arrests “have no impact on the city’s day to day operations.”

Mr. Molina and his wife were arraigned late Thursday morning in Edinburg, a city of 90,000 residents that is next door to McAllen and home to a University of Texas Rio Grande Valley campus. By the afternoon, the mayor was back at City Hall at a previously scheduled meeting. He sat in the council chambers, looking relaxed and checking his phone occasionally, while listening to presentations on the future use of the Ebony Hills Golf Course.

Mr. Paxton’s election fraud unit has become one of the busiest in the nation. It prosecuted 33 defendants in fiscal year 2018, and there are currently 75 active investigations statewide.

In January, the Texas Secretary of State’s office said it had discovered that about 95,000 people identified as noncitizens had matching voter registration records in the state, and that about 58,000 of them had voted in one or more Texas elections. David Whitley, the secretary of state, provided the information to Mr. Paxton for potential prosecution.