The Bluebirds were roared to victory against Manchester City by a capacity crowd in the first Premier League game in the Welsh capital last Sunday.
"The atmosphere on Sunday was incredible, probably the best I've played in, in my career I think," he told the club's official website.
"The fans have been tremendous ever since I've been here. They were fantastic last season and pushed us on to the Championship title.
"We'll need them again this campaign."
The Toffees are the first visitors for a 3pm kickoff at the Cardiff City Stadium and Connolly said that he and his teammates have another battle on their hands.
"Whenever you play Everton you know that they're a side who have got good players and a good team," he added. "They have a new manager so may play a different style. It'll be a tough physical game but we're ready for the challenge.
"We'll always work hard. I think that's one of our strongest assets and when we get the ball we'll go out to get the goals we need to win the game."
Connolly joined Cardiff from Queens Park Rangers last summer.