Claudio Ranieri's side beat Crystal Palace 1-0 on Saturday and are top of the Premier League by five points from Tottenham, who beat Bournemouth 3-0 on Sunday, and have the easier of the two run-ins with seven games remaining.
However, Sky Sports pundit Quinn says the nerves could hit home if they slip up just once, as they chase a first-ever top flight title having been 5,000/1 for the crown back in August.
Quinn told Super Sunday: "I watched the game live and it was up and down. It was unbelievable. You didn't know who was going to get the breakthrough. In absolute truth Crystal Palace could have won the game.
"You can be sure of one thing, they will go out and try and score goals. When they start to get picked off - if it happens - then we'll see what Ranieri has to do.
"He's just letting it flow at the moment, but when that one result goes wrong we'll know a lot more.
"All day long, Spurs have a tougher run-in, but I'm not so certain it's a matter of who you're playing against. It's a matter of how you deal with this incredible challenge that's ahead of you.
"I'm just convinced that Leicester are going to hit some kind of banana skin. I hope it doesn't happen and I'm with everybody else on that. I want Leicester to win the title.
Niall Quinn "I'm just convinced that Leicester are going to hit some kind of banana skin."
"But that's the time we'll see what they are really made of. I really hope they win it.
"I think there are games there that Spurs could trip up in. Leicester, on paper, have an easier run-in, but that goes out the window with what is at stake."
Tottenham are looking for their first title since 1961, and Graeme Souness believes their young side will push Leicester to the wire.
Barclays Prem
1 Leicester 66
2 Tottenham 61
3 Arsenal 55
4 Man City 51
5 West Ham 50
6 Man Utd 50
7 So'ton 47
8 Stoke 46
9 Liverpool 44
10 Chelsea 41
Souness echoed Quinn's belief that one defeat could rock Leicester, but says a Foxes title win would be great for the game.
"I want Leicester to win the league. I think it would be great for football. There is still a doubt in my mind: will they choke?
"I just think the nearer they get to the finishing line… I've got it in my head, maybe because it's never been done before.
"We shall see, but you have to say what a job the manager has done and what a job the players have done.
"Tottenham are a team that weren't fancied and all of a sudden they have come into the race with no one fancying them. They are not talking about winning it and will be delighted with Champions League football.
"Tottenham have got a youthful group that will keep going. I think they will push them all the way.
"If Leicester slip up once then you will see those nerves jangling."