The Hammers have not won a piece of silverware since the now-defunct Intertoto Cup in 1999, while their last major domestic trophy came in the form of the League Cup in 1981.
And Cole, who joined the east Londoners for a second spell in January, hopes to first improve on the club's 10th-place Premier League finish last season before eyeing a potential Wembley triumph.
|WEST HAM LATEST
|40/1||West Ham are 40/1 with BetVictor to win the Capital One Cup
"But for me personally, I would love to finish better than 10th, but equally I'd love to take this club to Wembley. Because at the end of the day, they are the days you remember in your career - it would be a massive achievement.
"West Ham are a big club but have been starved of success. You look at where we are in London and people want to come here and play in London and play for West Ham."
With the new season leading up to the 2014 World Cup, Cole acknowledged that Roy Hodgson will be assessing eligible players for the England squad heading to Brazil.
The 31-year-old has not featured for the Three Lions since the 2010 World Cup and is not confident of a call-up, but pointed to his experience on the international stage.
"The ambition will always burn every time you go out on the football pitch and you want to do well, if you're an Englishman and you're playing well you are inevitably going to be talked about.
"I have played over 50 times for my country so if I can get playing you never know but at the moment I'm just concentrating on West Ham."