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Edinburgh last qualified for the last eight of the tournament nine years ago when they faced four-times winners Toulouse in a quarter-final on French soil in 2004.
The Scottish capital club made it four wins from five in this season's competition courtesy of Phil Godman's "red zone" drop goal that clinched a 27-24 triumph against Racing Metro 92 and are now gearing themselves towards a contest in which they can make a real European impact.
Edinburgh's place in the Amlin Challenge Cup quarter final is already secured but a win will guarantee they reach the knockout stages of European rugby's premier competition for just the second time in their history.
"We need every Scottish rugby fan's backing," said Laidlaw.
"The Racing Metro game was huge and we went over to France and did a job. Now we're back home with a chance to achieve something special in front of our home support on Sunday at Murrayfield, with a lot of people backing us to the quarter-finals.
"With the crucial encounter beckoning, the club is urging all rugby supporters to back the club to surpass its Heineken Cup attendance record, set against London Wasps in 2008 (7,711), in its most important match in almost a decade."
Last month Edinburgh Rugby topped the Scottish PRO12 attendance record when 13,246 fans turned out at Murrayfield for the 1872 Cup derby with Glasgow Warriors.
