Did Mexico have an Austrian emperor?

Yes, Mexico had an Austrian emperor. Maximilian I of Mexico, also known as Ferdinand Maximilian Joseph, was the emperor of Mexico from 1864 to 1867. He was born in Vienna, Austria, and was a member of the House of Habsburg-Lorraine. Maximilian was invited to become emperor of Mexico by a group of Mexican conservatives who were opposed to the liberal president, Benito Juárez. He accepted the offer and arrived in Mexico in 1864 with a French army to support him. Maximilian's reign was short-lived, however. He was defeated by Juárez's forces in 1867 and was executed.