Bedford host Leinster A, English Championship table-toppers Worcester Warriors entertain Moseley and Bristol meet Ayr in the other quarter-finals.
Pontypridd were joint top scorers acorss the four Pools with 23, along with Mike Rayer's Bedford, thanks tor wins over Llandvoery, Doncaster Knights, Ayr, Rotherham and Birmingham & Solihull. Llanelli were runners-up to Bristol in Pool B with four wins from five.
"We know Llanelli very well and we know how tough this game is going to be," said McIntosh.
"We have beaten them once already this season at Sardis Road (19 v 13 in the Premiership) but they had a depleted team and will be much stronger this time.
"We have played some of our best rugby in this competition and we dipped our feet in the water last year and got a taste for it.
"When we saw how the fixtures fell this year we thought this was a competition we could really go for.
"Everyone at the club, the players, coaches and supporters, are excited by the tournament, it brings a new dimension to the season, playing against teams and players you know very little about."