Is the word quê the same in Brazilian and European Portuguese?

Yes

The word "quê" (with the circumflex accent) is functionally identical in both Brazilian and Continental Portuguese. There are no differences in meaning, grammar, spelling, or the orthographic rules governing its use.

The accent is applied only when the word is "tonic" (stressed). This occurs in three specific linguistic contexts:

1. At the end of a sentence or clause: When the word follows a pause or ends the thought (e.g., "Você quer o quê?" / "What do you want?").
2. As a noun: When it is used to mean "a certain quality," "something," or "a hint of something" (e.g., "um quê de mistério" / "a hint of mystery").
3. In isolation: When used as a standalone interjection (e.g., "O quê?" / "What?").

The only differences between the two dialects regarding this word are the phonetic realization of the vowel and the surrounding syntax and vocabulary used in the sentence.

Brazilian Portuguese Examples

1. Você está fazendo o quê? (What are you doing?)
2. Ele tem um quê de maldade no olhar. (He has a hint of malice in his gaze.)
3. O quê? Você não entendeu nada? (What? You didn't understand anything?)
4. Você comprou o quê na padaria? (What did you buy at the bakery?)
5. Esse perfume tem um quê de baunilha. (This perfume has a hint of vanilla.)

Continental Portuguese Examples

1. Estás a fazer o quê? (What are you doing?)
2. Ele tem um quê de maldade no olhar. (He has a hint of malice in his gaze.)
3. O quê? Não percebeste nada? (What? You didn't understand anything?)
4. Compraste o quê na pastelaria? (What did you buy at the bakery?)
5. Este perfume tem um quê de baunilha. (This perfume has a hint of vanilla.)