In American English, it is exam center, and in British English, it is exam centre. That simply reflects how the word center/centre is spelled in the two different variants of the language. This difference is also seen in word pairs such as theater/theatre and meter/metre.
Center/centre is a noun that refers to the literal or figurative middle of something or, as in the case of “exam centre/centre,” to a building that serves a particular purpose.
As a verb, center/centre means “place or be in the middle or at the heart of something.”
You can use the QuillBot Grammar Checker to ensure you use center or centre correctly.