The club has been without a major trophy for over 30 years, with their last significant win being the FA Cup in 1995.
With a Carabao Cup third-round tie against Wolverhampton Wanderers approaching, Tarkowski believes his team has a good chance of lifting silverware this season.
Although Everton has won trophies across nine different decades, they have yet to win the EFL Cup.
Tarkowski takes inspiration from recent cup wins by Crystal Palace and Newcastle United, and hopes to follow their example with his team.
"We're focusing on doing well in the cups this year, and after winning the first round, we want to go further," Tarkowski said.
"We've seen teams outside the top six win domestic trophies, so there's no reason why we can't do the same," he added.
Tarkowski emphasized that the team is committed to winning a trophy and going as far as possible in the competition.
"Every season, there's an emphasis on the cup, but this year it's been a specific focus among the players and staff to win something," he said.
Everton has a rich history of success, with nine league titles, five FA Cups, one European Cup Winners' Cup, and nine Charity Shields.
This season, under manager David Moyes, they have a chance to add to their trophy collection and end their drought.
Their first major test comes against Wolves in the Carabao Cup, a crucial step towards potentially securing a much-needed trophy.