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

Yes.

The word "sugerem" is identical in meaning, grammar, and spelling in both Brazilian and Continental Portuguese. The only difference lies in pronunciation. In Brazilian Portuguese, the nasal diphthong "-em" at the end of the word is often pronounced with a more open, semi-vocalized sound (approaching an "ey" sound in many dialects), whereas in Continental Portuguese, the nasalization is typically more closed and distinct.

Brazilian Portuguese Examples:

1. Eles sugerem que a gente vá ao shopping. (They suggest that we go to the mall.)
2. Os médicos sugerem um remédio para a dor. (The doctors suggest a medicine for the pain.)
3. Eles sugerem que a gente mude o plano. (They suggest that we change the plan.)
4. Os professores sugerem um lanche rápido. (The teachers suggest a quick snack.)
5. Eles sugerem que a gente descanse um pouco. (They suggest that we rest a little.)

Continental Portuguese Examples:

1. Eles sugerem que nós vamos ao centro comercial. (They suggest that we go to the shopping center.)
2. Os médicos sugerem um medicamento para a dor. (The doctors suggest a medication for the pain.)
3. Eles sugerem que nós alteremos o plano. (They suggest that we alter the plan.)
4. Os professores sugerem um lanche ligeiro. (The teachers suggest a light snack.)
5. Eles sugerem que nós descansemos um pouco. (They suggest that we rest a little.)