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

Yes.

The word "quer" is the third-person singular present indicative of the verb querer (to want). There are no differences in its meaning, grammar, or spelling between Brazilian and Continental Portuguese. The only differences are in pronunciation: Brazilian Portuguese typically features more open vowels and a variety of "r" sounds (ranging from retroflex to aspirated), whereas Continental Portuguese tends to have more closed or reduced vowels and a more distinctly tapped or alveolar "r."

Brazilian Portuguese Examples

1. Ele quer um café. (He wants a coffee.)
2. Ela quer o celular. (She wants the cell phone.)
3. Ele quer um sorvete. (He wants an ice cream.)
4. Ele quer pegar o ônibus. (He wants to catch the bus.)
5. Ele quer o trem. (He wants the train.)

Continental Portuguese Examples

1. Ele quer um café. (He wants a coffee.)
2. Ela quer o telemóvel. (She wants the mobile phone.)
3. Ele quer um gelado. (He wants an ice cream.)
4. Ele quer apanhar o autocarro. (He wants to catch the bus.)
5. Ele quer o comboio. (He wants the train.)