Arsene Wenger's side have not won a competition since the 2005 FA Cup and have already been dumped out of the Capital One Cup this term, while they lie a distant sixth in the Premier League and face a daunting Champions League last-16 clash with Bayern Munich next month.
The Gunners must come through a third-round replay with Swansea to book an FA Cup fourth-round trip to Brighton, after being held to a 2-2 draw in Wales on Sunday, but Wilshere and his team-mates are desperate to finally taste success.
"We all want a trophy," said the 21-year-old. "As players, when you finish your careers you look back and see how many trophies you have won, so we all want trophies, and the fans want trophies as well.
"We are hungry for trophies and if we get a trophy this year, then maybe it can open the way for a few more.
"We have got a good group of young players here and experienced players as well.
"So, if we get a trophy this year or in the next couple of years, we can really build something over the next few years.
"Obviously it is easy to sit here and say that when we get one trophy it will happen, but we need to get that first trophy. We know how tough the FA Cup is this year and we are still in the Champions League as well.
"So we are in a good position, we just need to make sure we get the results now."
Either side of the replay against Swansea, Arsenal face defending Premier League champions Manchester City and European champions Chelsea as they look to secure a top-four spot.
But Wilshere does not believe the next few games are make-or-break for Arsenal's season.
He said: "We have got a tough few games but we know in England that this time of the year is so important. We've got a big game on Sunday and we know City are a good team. But we are at our home and we have to take the game to them.
"I'm not sure about make-or-break. We are not in the position in the league where we want to be. We want to be higher. We have a game in hand and we have to make sure that we make it count."