A trip south to take on the Pirates is their reward for qualification with coach Geraint Lewis looking forward to the tie.
"It's a tremendous achievement to qualify," he said.
"We've come through it and it's a testament to the players here who are determined and put a lot of effort in. We're now looking forward to a quarter final."
Pontypridd coach Paul John added: "Any draw at this stage of the competition was going to be extremely tough and we have a real challenge on our hands travelling away to take on Cornish Pirates.
"They are a strong team who are more than holding their own in the English Championship. We know from our encounters with London Welsh and London Scottish in the pool games how much quality there is amongst the professional English clubs.
"This will be a fresh challenge for us as we have not played the Pirates as yet and we will travel to Penzance with every intention of putting in a big performance and coming away with a win.
"Our supporters are I'm sure already relishing the trip to Cornwall. It will be another big day out for them which will be no more than they deserve after following us through thick and thin with great enthusiasm so far this season."
The other quarter final fixtures see Bristol host Rotherham in an all English Championship affair. On a similar note, Leinster A and Munster A are set to clash in the last eight while in the final fixture, Plymouth Albion face a long trip north to face Leeds Carnegie.
If Pontypridd get the better of the Cornish Pirates, they will play the winner of the meeting between Leinster and Munster in the last four.
British & Irish Cup quarter final fixtures: Ties to be played April 4,5,6:
Match 1: Bristol Rugby v Rotherham Titans
Match 2: Leinster 'A' v Munster 'A'
Match 3: Cornish Pirates v Pontypridd
Match 4: Leeds Carnegie v Plymouth Albion
British & Irish Cup semi final fixtures: Ties to be played May 2/3/4:
Winner of Match 3 v Winner of Match 2
Winner of Match 1 v Winner of Match 4