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

Yes

The word "quarto" is identical in both Brazilian and Continental Portuguese regarding its meaning (room, fourth, or a quarter of something), its spelling, and its grammatical function. The only difference is in pronunciation. In Brazilian Portuguese, the vowels are generally more open and the rhythm is more melodic and syllable-timed. In Continental Portuguese, unstressed vowels are often reduced or almost silent, making the word sound much more "closed" and consonant-heavy.

Brazilian Portuguese Examples:

1. Meu quarto é muito pequeno. (My room is very small.)
2. Eu estou arrumando o meu quarto. (I am tidying my room.)
3. O quarto de visitas está pronto. (The guest room is ready.)
4. Ele ficou no quarto o dia todo. (He stayed in the room all day.)
5. O quarto está muito bagunçado. (The room is very messy.)

Portuguese (Portugal) Examples:

1. O meu quarto é muito pequeno. (My room is very small.)
2. Eu estou a arrumar o meu quarto. (I am tidying my room.)
3. O quarto de visitas está pronto. (The guest room is ready.)
4. Ele ficou no quarto o dia todo. (He stayed in the room all day.)
5. O quarto está muito desarrumado. (The room is very messy.)