Barca beat Santos to lift the trophy at the weekend but Real manager Mourinho appeared to scoff at his rivals' achievements, offering sarcastic congratulations for winning "two little games" and stating he would prefer to win the Champions League.
Alves responded by saying: "I always say Mourinho plays his role carefully and tries to devalue what others win, only giving value to what he earns. Envy is very bad."
That prompted Pepe to become involved and after Real booked their place in the last 16 of the Copa del Rey with a 5-1 win over Ponferradina on Wednesday, he spoke to Marca TV's 'Futboleros' programme.
The Brazilian said: "Envy? We have no envy of anyone. I play for the best team in the world. Real Madrid is the biggest club in the world and cannot be envious of others."
Of Wednesday's game, the defender added: "We played with humility, respect and worked well. All Madrid is happy people, we will spend a quiet Christmas and in January come back to work for everything. "
Pepe was also honoured to have led Real for the first time and said: "I am very happy to have been captain and win. This is the best club in the world and is a great joy.
"We have to keep working well to celebrate something at the end of the season. "