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

No.

While the meaning, spelling, and grammar of "telefonar" are identical in both dialects, there is a difference in typical usage and register. In Brazil, "telefonar" is considered formal, professional, or slightly old-fashioned; in everyday conversation, Brazilians almost exclusively use the verb "ligar." In Portugal, "telefonar" is much more common and is used in standard, everyday speech, though "ligar" is also frequently used.

Brazilian Portuguese (Using "telefonar" in its natural, formal context)

1. Vou telefonar para o escritório para confirmar a reunião. (I am going to call the office to confirm the meeting.)

2. Ele telefonou para o advogado ontem à tarde. (He called the lawyer yesterday afternoon.)

3. Por favor, telephone para o suporte técnico assim que puder. (Please, call technical support as soon as you can.)

4. Ela precisa telefonar para o cliente para tratar do contrato. (She needs to call the client to handle the contract.)

5. Nós telefonamos para a recepção para reservar o quarto. (We called the reception to reserve the room.)

European Portuguese (Using "ligar" as it is the most natural way to express the same idea)

1. Vou ligar à minha mãe agora. (I am going to call my mother now.)
2. Podes ligar ao médico para marcar uma consulta? (Can you call the doctor to make an appointment?)
3. Ele ligou-me para dizer que ia chegar tarde. (He called me to say he was going to arrive late.)
4. Ela tem de ligar ao chefe para avisar do problema. (She has to call the boss to warn [him] about the problem.)
5. Ligaram ao restaurante para saber se havia mesa. (They called the restaurant to see if there was a table.)