Number 1 seeds Bedford have been drawn against Llanelli, the sole Welsh representatives left in the competition, while Newcastle Falcons play Nottingham in the only all-English tie.
2011 winners Bristol are at home to Leinster 'A' with Cornish Pirates and reigning champions Munster 'A' facing a repeat of the inaugural final, won by the Pirates, in 2010.
Bedford finished 19 points clear of their nearest rivals Stirling at the top of Pool 2 after going through the pool stages without dropping a single point.
Bottom seeds Llanelli got through at the expense of Moseley on points difference. Llanelli managed to hold Bedford to a 25-25 draw on their last visit to Goldington Road in the British & Irish Cup in December 2011.
Bedford, who lost out to Bristol in the 2011 British & Irish Cup final, are searching for their first silverware since lifting the Powergen Shield at Twickenham in 2005.
Newcastle were afforded second seed status ahead of Bristol on points difference to set up a Kingston Park showdown with Nottingham. The Green & Whites defied all the odds to make it through to the last-eight.
Having lost their first two games in the competition Nottingham went into their final round match against Llandovery at Meadow Lane needing a bonus point win, a clear 10-point winning margin, and to deny the Welsh Premiership outfit a bonus point in order to progress. Two tries from Andy Savage helped them achieve all three goals in a 31-15 win.
Like Newcastle, Bristol finished with 27 points from the pool stages after a final round 61-12 win over Cardiff.
Going into the tie Bristol needed just one point to be guaranteed a place in the last-eight, while also knowing that a win would secure them a home tie.
They achieved a sixth straight British & Irish Cup win in style, with seven different players scoring tries including a double each for George Watkins and Ben Mosses, while Tristan Roberts was successful with eight out of his nine conversion attempts.
A record of five wins and one defeat was enough for Cornish Pirates to top Pool 3 ahead of Carmarthen, Swansea and Dundee.
Munster 'A' won five and drew one of their six Pool 7 games, scoring just 123 points in the process - the lowest of any of the sides in the quarter-final draw.
Cornish Pirates and Munster 'A' have played each other once since the 2010 final, the Pirates achieving a memorable 10-6 pool win in Clonmel in November 2011.
BRITISH & IRISH CUP FINAL SEEDINGS:
1. Bedford Blues (30 points), 2. Newcastle Falcons (27*), 3. Bristol (27), 4. Cornish Pirates (24), 5. MUNSTER 'A' (23), 6. LEINSTER 'A' (22), 7. Nottingham (20), 8. Llanelli (19)
* Superior scoring difference
BRITISH & IRISH CUP QUARTER-FINALS:
Friday, April 5 -
Newcastle Falcons v Nottingham, Kingston Park, 8pm
Saturday, April 6 -
Bedford Blues v Llanelli, Goldington Road, 3pm
Sunday, April 7 -
Cornish Pirates v MUNSTER 'A', Mennaye Field, 3pm
Bristol v LEINSTER 'A', Memorial Stadium, 3pm