Victories at Bourgoin and home to Bath Rugby have given Treviso a 50 per cent return but Green is adamant that they are not finished yet.
"We have to consolidate on what we have done so far and I am pleased with the two wins," said the former All Blacks wing.
"Now we have to beat Bourgoin and do the double over them. At the same time we will see how the other results fall and see where we go from there.
"With Leinster seven points clear at the top of the group they are obviously clear favourites for automatic quarter-final qualification but we are not entirely out of it and the Bourgoin match is a huge one for both the club and Italian rugby.
"The boys are very positive about both remaining games and when we go to Dublin in Round 6 we will make sure we enjoy ourselves and have a good crack at Leinster."
Treviso - who have failed to qualify for just two of the 10 Heineken Cup tournaments - enjoyed their best season in Europe so far when they won three of their Pool matches in 1998 / 1999.
Overall, NEC Harlequins (twice), London Wasps, Bath Rugby, Pontypridd (twice), Bourgoin (twice), Llanelli, Edinburgh, Caledonia, Glasgow Caledonians and Farul Constanta have been their 13 Euro scalps to date.
"Sometimes people get a bit unrealistic about what we can achieve in the Heineken Cup," said Green.
"But we have played well in the tournament this season and that has produced good publicity with plenty of nice things said about Treviso and Italian rugby.
"We accept that, as the top Italian club, if we don't do well then it is unrealistic to think that any other Italian club can really make a mark though any win - such as Parma's European Challenge Cup win against Saracens - has to be good for the game in Italy.
"The players have had a decent break - although we had nine players with the national squad for their four-day camp - and aside of the long term injuries I hope we will have a full strength squad for the Bourgoin match," said Green.
"As we fly the Italian flag in Europe in the Heineken Cup, the pressure is on us to perform in the final two rounds of the tournament."
Laurent Seigne, the Bourgoin head coach whose side have picked up just one losing bonus point so far, said: "When you've won the Heineken Cup like me with Brive in 1997, losing 34-0 home is like being wounded.
"For the coming match against Treviso, of course we'll have in our hearts the motivation to clear our honour and the score-board!"