Coquelin was on loan at Freiburg when Arsenal won their first major trophy in almost a decade with a 3-2 victory over Hull City at Wembley in 2014.
However, the Frenchman was an integral part of the squad that defended the trophy last season, defeating Aston Villa 4-0 in the final, and he is now keen to add to his trophy cabinet.
"It was my first major trophy with the club so I don't want to sleep on this," he told the Arsenal Weekly podcast ahead of Saturday's third-round tie against Sunderland.
"I'm still hungry, even hungrier, and I want more, like everyone else at the club. The players are really stepping up and really want this silverware.
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"It was a great moment, but you need a little bit of the magic of the FA Cup to come back before the memories come back. When it does start, people will be looking at the pictures of themselves with the trophies and thinking that we need to hold it for a third time.
"What is nice is that the first round of the competition is at home and that's great for the fans. It's always going to be tough against Sunderland. They gave us a difficult game a couple of weeks ago, so we know it will be tough.
"They're going to want to beat the champions as well, so they will be really, really motivated and beating us could give them a boost in their season as well. Everyone will be prepared for it.
"I don't know if anyone's ever held the FA Cup three times in a row before, but it would mean something big for the club. We want to win any trophy so if we can get this one, we will."