Saints have won the last four meetings between the teams, which included last season's League Cup semi-final and Scottish Cup final.
Hibs did not score in any of those four defeats and only won one of the six encounters between the teams last season, thanks to a stoppage-time penalty in August.
"Our players are going with the belief that they can get a result," said Davidson, whose team reached the Premier Sports Cup semi-finals, along with Hibs, in midweek.
"That's a great thing to have. We know we can get a result if we perform well.
"It's not about getting into their heads, it's about getting into my players' heads that we can go there and do something.
"That's the biggest thing for me, the players realising that we can go to Easter Road and get something.
"Every game was pretty close, even the one they won at the start of the season.
"I am expecting a tight game again. I am expecting us to go there and make it really difficult for Hibs.
"They have a real good attacking threat and we have to make sure we stop that."
After Saints set up a Hampden clash with Celtic, Hibs won 3-1 at Tannadice to seal a fifth consecutive semi-final at Hampden, where they will take on Rangers.
"It will be a really tough game," Davidson said. "Hibs were very fortunate to manage to hold on to some of their top players, and also add to their squad.
"They are a very good team, I really enjoy watching them play. We are going to be up for the battle.
"They are very good on the counter-attack and have some very good players up front that can hurt you.
"They had a fantastic season with where they finished in the league and they have carried it on this season."