In Texas, a County Judge like this is different than elsewhere: https://en.wikipedia.org/wiki/County_judge
In Texas, there are two types of county judges: constitutional county judges, which the Texas Constitution requires each county to have, and statutory county judges (or county court at law judges), which the legislature has created for the more urban counties.
In rural counties, the constitutional county judge acts both like the mayor of the county and hears some court cases (mainly misdemeanor criminal, appeals from Justice of the Peace and small claims courts, and uncontested probates). In urban counties, such as Fort Bend, the constitutional county judge has no judicial duties and is just the county mayor (with the same vote as each of the four county commissioners), while county courts at law hear any judicial cases that would otherwise be heard by a county judge.