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

Yes

The word "pergunta" is identical in spelling, meaning, and grammatical function in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the vowels are generally more "open" and clearly articulated. In Continental Portuguese, there is a significant amount of "vowel reduction," meaning unstressed vowels are often shortened or nearly silent, making the word sound more "clipped" or "compressed" to a foreign ear.

Brazilian Portuguese Examples

1. Eu tenho uma pergunta para você. (I have a question for you.)
2. Ele sempre faz a mesma pergunta. (He always asks the same question.)
3. Posso te fazer uma pergunta rápida? (Can I ask you a quick question?)
4. A pergunta foi muito difícil de responder. (The question was very difficult to answer.)
5. Ela pergunta se você vai à festa. (She asks if you are going to the party.)

Continental Portuguese Examples

1. Tenho uma pergunta para ti. (I have a question for you.)
2. Ele faz sempre a mesma pergunta. (He always asks the same question.)
3. Posso fazer-te uma pergunta rápida? (Can I ask you a quick question?)
4. A pergunta foi muito difícil de responder. (The question was very difficult to answer.)
5. Ela pergunta se tu vais à festa. (She asks if you are going to the party.)