A much-changed Celtic side kept up the momentum by coming from three goals down to draw against Rapid Vienna in Austria on Thursday, with Fortune bagging a double.
The upturn in form, which followed two wins from 10 games, will be tested by the Jambos at Tynecastle in this weekend's Scottish Premier League clash.
But Grant feels the Hoops players can overcome the traditionally tricky fixture if they continue to keep faith in their ability.
"I can honestly say I've never played at Hearts at any time and had an easy game," Grant said.
"Tynecastle always has a fantastic atmosphere and intimidating in the respect they make it very difficult for you, very competitive.
"They have good quality players and they have shown that against us. It took a last-minute goal to win the league match and then obviously they put us out of the cup.
"We're going into the game knowing it's a tough, tough match for us.
"But we're playing well, and even in some of the games we were getting criticism, we thought we were playing well. We weren't a million miles away.
"But we know the expectations at this club, you have to play well and win, and that doesn't change.
"But the way the boys are playing, they have confidence and that's a big thing. If you believe in yourself you always have a big, big opportunity.
"People keep knocking your confidence but, as long as you don't lose it, you'll do okay, and I think the boys are proving that at the moment."
Hearts are facing a similar test of their belief after following their Co-operative Insurance Cup win at Celtic Park with one win from six matches.
Csaba Laszlo's side are rooted in the bottom half of the SPL after failing to build on their third-place finish last term.
And Zaliukas, who is looking to make his comeback after almost three months out with an ankle injury, believes they have taken time to adapt following a summer turnaround in players.
"The players we had last year played here a couple of years so maybe we had a better understanding," Zaliukas said.
Nevertheless, the Lithuanian would be happy to see more new recruits in January.
"If we get a couple of players - strong players, obviously - maybe results will be better," the defender said.
Mowbray is likely to bring back several first-choice players but there is a doubt over in-form striker Georgios Samaras (back) and left-back Danny Fox (hamstring).