Cockerill’s side sealed a home tie in the last eight when they beat Montpellier 19-10 on Friday to retain top spot in Pool 5.
They will host Munster on the final weekend of March while Glasgow face a return trip to Saracens following their defeat in London on Saturday.
Edinburgh have won all 11 home games this season and Cockerill feels they have earned the right to attract a huge support after more than 11,000 watched them beat the French side.
Cockerill said: “Seventy-five per cent of teams win their quarter-final when they play at home.
“Hopefully we get a bigger crowd. Financially it’s great for us. It makes the world go round.
“We want to have 30, 40, 50 thousand people here for a quarter-final. Why not? Because we actually have a team that’s worth watching.
“When I came here, why would you come and watch because we were pretty c***. But now we are not, we have shown we are a bloody good side.
“Come and support us because the boys deserve it.”
Glasgow lost 38-19 to Saracens in their group decider but Edinburgh’s win had already ensured they would finish as one of the three best runners-up at worst.
But they can be encouraged ahead of their return visit by the fact they scored three first-half tries before Saracens streaked clear to make it a harsh scoreline in the second half.
Warriors head coach Dave Rennie said before their quarter-final opponents were confirmed: “With a bit of luck we’ll have some of our injured players back if we come here for the quarters. We’d bring a fair bit of steel.
“It’s nice to know we’re still alive. If it’s here that we’re coming back to - which is likely - we’ll give it a good lick.
“We fell short today though. Saracens are a good side so you’ve got to get them out of their comfort zone, which we tried to do.”
Edinburgh’s win sealed an historic achievement for Scottish rugby, with two representatives in Europe’s top eight for the first time, but there were plenty of negatives ahead of the Guinness Six Nations.
Edinburgh and Scotland flanker Hamish Watson suffered a suspected broken hand while three members of Gregor Townsend’s squad sustained shoulder injuries against Saracens: Jonny Gray, Ryan Wilson and Grant Stewart.
Rennie told BBC Radio Scotland on Sunday: “They will all be screened over the next few days to assess the severity of it, but hopefully not too bad.”
Another Scotland player, centre Sam Johnson, suffered a head injury.
Full draw: Edinburgh v Munster
Leinster v Ulster
Racing 92 v Toulouse
Saracens v Glasgow Warriors
The ties played over the final weekend in March.