“Apparently she had to either pay the fine on the spot or go see a judge immediately to contest it.”
The ticket was $126. She has the mañana mentality and refused to deal with it that very day. This is small town New Mexico and they want the $126 fine money. So you don’t get to “mail it in” later. You pay now or see the local yokel judge and maybe get lucky with him.
She made up her own third option which was to do neither and drive away.
“Pay now or contest it with a judge immedidiately” is typical of robber baron one-horse towns abusing the law. If they catch you at night or on the weekend and the court is not in session your choice is “pay now or sit in jail until the judge can see you.” Evil.