Zebre, on paper at least, should offer little threat to the Londoners, either in Italy on Saturday or when the teams meet at Twickenham Stoop the following week.
Quins, though, saw their hopes of reaching last season's knockout phase wrecked by a 9-8 loss against Connacht in Galway.
"We underachieved in the Heineken Cup last season," Quins wing Tom Williams said.
"Connacht did brilliantly to beat us in the final pool game, and that cost us, so we know we cannot afford to take any side lightly.
"Zebre have only played a few games together. They've played twice in the Heineken Cup and then they have had their league fixtures in the RaboDirect PRO12, so we've done as much research as we can.
"We are doing well in all the competitions - we are top of the Premiership and looking good in the Heineken Cup - but it is a sign of how far we have come and developed as a team that we are still not satisfied."
Three of Quins' Premiership rivals - Saracens, Exeter and Sale Sharks - will also be in European action on Saturdeay, with Saracens visiting Munster, Chiefs heading to the Scarlets and Sharks entertaining Jonny Wilkinson's Toulon.
But one English challenger, Northampton, potentially face a tougher task than all of them when Pool Four rivals Ulster arrive at Franklin's Gardens on Friday evening.
Ulster are the only remaining unbeaten team in European rugby's leading three domestic leagues this season - the Premiership, RaboDirect PRO12 and the French Top 14.
"Ulster have a strong winning habit this season and are coming to Franklin's Gardens in the best possible form," Saints rugby director Jim Mallinder said.
"Their squad is full of talented players who, time and again this season, have shown a never-say-die attitude and the ability to get the scores when they needed them.
"It is going to be a massive challenge for us, we know that. But training has been very positive this week and we are coming into the game on the back of three successive wins ourselves."
England flanker Tom Wood returns from autumn Test duty to pack down at openside for Northampton, while full-back Ben Foden also features after making a successful return from injury against Premiership opponents Sale last Friday.
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