The 22-year-old made his international debut against Wales in the 2009 Six Nations and signed for Aironi from Italian Super 10 outfit Rugby Parma this summer.
He will now begin his rehabilitation programme and it is hoped he will be fit in time for the start of pre-season training next summer.
Aironi lost Quartaroli's centre partner, South African Dylan Des Fountain, in the same match with the same injury.
Des Fountain has also started his rehabilitation but will not play again until next season.
New Zealander Rodd Penney, who signed from Saracens to cover the Italian side's midfield woes, will make his Magners League debut against Edinburgh tomorrow.
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