Terim, who is in his third spell in charge of the Turkish side, knocked out German club Schalke in the last round and is positive that his side can cause an upset in the last eight.
"I trust in my team and in my players. We came from the group of Manchester United and then knocked out Schalke, so we fought very hard and we will continue to fight," Terim told reporters.
"We will look to continue doing our best. We are going to play with the idea of pushing every limit.
"I have never told my players that one goal was enough. I always tell them to overcome their limits, to keep on growing."
The 59-year-old admitted that his side have drawn one of the toughest teams left in the competition in Jose Mourinho's side, but refused to use it as an excuse for them not to progress.
He added: "We have got the most difficult opponents in the quarter-finals. This is not an excuse though.
"If I were the coach of any of the other seven teams I would also have asked to be drawn against Galatasaray.
"We must analyse what our team has done in this tournament up until now though.
"Me and Mr Mourinho have a long friendship. He is a very open and modest person. We have been speaking about things that two good friends speak about.
"When the draw was made we drew a team whose goal is to win the title. Madrid have some of the most important players in the world and they have maybe the best coach in the world.
"Madrid at home are very effective. They have achieved great success in this stadium. They have one of the most dangerous and powerful players in the world in [Cristiano] Ronaldo."