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

Yes

The word "quase" is identical in both Brazilian and European (Continental) Portuguese. There are no differences in meaning, grammar, or spelling. The only difference lies in pronunciation: in Brazilian Portuguese, the vowels tend to be more open and clearly articulated, whereas in Continental Portuguese, unstressed vowels are often more closed or even reduced/elided.

Brazilian Portuguese Examples

1. Eu quase dormi no ônibus. (I almost fell asleep on the bus.)
2. Já é quase meio-dia. (It is almost noon.)
3. A comida quase queimou. (The food almost burned.)
4. Ele quase chegou aqui. (He almost arrived here.)
5. Quase não deu tempo de comer. (There was almost no time to eat.)

Continental Portuguese Examples

1. Eu quase adormeci no autocarro. (I almost fell asleep on the bus.)
2. É quase meio-dia. (It is almost noon.)
3. A comida quase ardeu. (The food almost burned.)
4. Ele quase chegou aqui. (He almost arrived here.)
5. Quase não houve tempo para comer. (There was almost no time to eat.)