Is the word no the same in Brazilian and European Portuguese?

Yes

The only difference is in pronunciation. In Brazilian Portuguese, the nasal diphthong in "não" (the word for "no/not") is typically pronounced with a more open and resonant nasal vowel. In Continental Portuguese, the pronunciation tends to be more closed, and the nasal sound can be more compressed or even neutralized in rapid speech. The spelling, meaning, and grammatical function are identical in both dialects.

Brazilian Portuguese Examples

1. Eu não sei o que dizer. (I don't know what to say.)
2. Você não vai à festa? (Aren't you going to the party?)
3. Não tem problema nenhum. (There is no problem at all.)
4. Não faz mal. (It doesn't matter / No harm done.)
5. Você não viu o cachorro? (Didn't you see the dog?)

Portuguese (Portugal) Examples

1. Eu não sei o que dizer. (I don't know what to say.)
2. Tu não vais à festa? (Aren't you going to the party?)
3. Não há problema nenhum. (There is no problem at all.)
4. Não faz mal. (It doesn't matter / No harm done.)
5. Tu não viste o cão? (Didn't you see the dog?)