Neil Lennon's side were instead handed a home tie against St Johnstone, while Rangers will face Inverness Caledonian Thistle at Ibrox.
In the other two last-eight ties, Aberdeen host St Mirren while Hearts travel to face Dundee United. The match at Tannadice will be televised by the BBC on October 31 at 7.15, the others will be played on either the 30th or 31st.
With no prospect, in theory, of an Old Firm league match for at least three seasons, anticipation was high that the pair would be drawn to face each other in the cup. Ledley, however, believes that Celtic's involvement in the Champions League this season more than makes up for the loss of the Glasgow derby.
"Most people wanted to see an Old Firm game against Rangers but as players we don't care," the Welshman said. "Am I missing the Old Firm games? No, nowhere near it. When you're playing in the Champions League it makes up for it. It was a fantastic achievement what we did in Moscow [beating Spartak 3-2 on Tuesday evening]. We are just happy with the way we are playing and who we are playing against.
"It doesn't bother us at all [if we face Rangers this season]. We just want to keep playing and see what happens. Our main aim is to win the league, win all the cups and to get as far as we possibly can in Europe. Those are our main aims. Rangers are not in the SPL so we have to get on with things. Can we do the clean sweep this season? Fingers crossed, yes, that is definitely our aim. It's going to be difficult because people will be wanting to [beat us] but our aim is to win all three trophies."
Neil Alexander, the Rangers goalkeeper, admitted there had been some talk in the Ibrox dressing room beforehand about the prospect of a match-up with Celtic. Now he hopes the pair go on to meet in the semi-final or the final. "There was a bit of joking among the boys," he said. "It was always a possibility. From the media and the fans' point of view it would have been fantastic but hopefully we can meet them in further rounds if we can beat Inverness."
Victory over Motherwell, the then Clydesdale Bank Premier League leaders, in the last round has imbued Rangers with confidence that they can go further. "We're not going to get carried away but the Motherwell game was fantastic," Alexander added. "Everything clicked and it showed how good we can be. The experienced boys who have been brought up at Rangers know that we go into every game and every cup to win. A draw is a catastrophe and a defeat is even worse. It's win or nothing at Rangers and we try to bring that mentality into each game."
Frazer Wright, the St Johnstone defender, hoped recent victories over Celtic would stand them in good stead for their tie at Parkhead. "It is probably the toughest draw of the round but it is done now, we just need to go there and try to get a result.
"Beating Celtic [earlier this season] put us on a good wee run and last year was the same. We went to Parkhead without having won a game and then won and went on a run so we know we can do it. We just need to have belief."
Danny Lennon, the manager of St Mirren, conceded a trip to Pittodrie in the last eight would be tough for his side. "We're delighted to be in it and it's our fourth quarter final in succession. There are no easy games at this level and we've got a difficult one away to Aberdeen. But we've done well up there in the past and got a draw the last time.
"We saw what Kilmarnock did last season [winning the trophy] and we would love to give our wonderful fans something similar if we could."