Favourites Barca booked their place in the last eight thanks to their 4-0 victory over AC Milan in the second leg of their last-16 tie on Tuesday to complete a 4-2 aggregate success.
Big-spending French side PSG progressed after seeing off Valencia 3-2 on aggregate, although Beckham was only an unused substitute in the second leg.
Paris St Germain sports director Leonardo said: "We are going to play against the best team. It is not easy but no draw would have been easy anyway. Paris deserves to be part of the big party."
The possibility remains of a clash of the Spanish giants in the final at Wembley on May 25 after Real Madrid were paired with Galatasaray in the quarter-finals, a game that should see Real boss Jose Mourinho lock horns with former Chelsea striker Didier Drogba, who now plays for the Turkish club.
Spain's third representatives, Malaga, have been drawn against Borussia Dortmund, while last year's beaten finalists Bayern Munich, who stuttered past Arsenal on Wednesday, will take on Celtic's conquerors Juventus.
Bayern chief executive Karl-Heinz Rummenigge said: "That is certainly not a dream draw for us.
"Statistically we have not had good experiences with Juventus. It will be difficult and we will need to have two good days against them to reach the semis."
Borrussia Dortmund sports director Michael Zorc was content with the draw after the Bundesliga champions were paired with Malaga.
"We can live with this draw," he said. "At least it is better than playing Barcelona, Real or Bayern."
The Premier League does not have a team in the Champions League quarter-finals for the first time since the 1995-96 season.
Champions League quarter-final draw: Malaga v Borussia Dortmund, Real Madrid v Galatasaray, Paris Saint-Germain v Barcelona, Bayern Munich v Juventus.
PSG v Barcelona
Bayern Munich v Juventus
Malaga v Borussia Dortmund
Real Madrid v Galatasaray
Borussia Dortmund v Malaga
Galatasaray v Real Madrid
Barcelona v PSG
Juventus v Bayern Munich