Gronbaek sustained a muscle tear in his right thigh during Denmark's 3-2 Nations League defeat by Norway on Thursday.
The 25-year-old had started all five of Hamburg's competitive matches this season under Merlin Polzin.
He had made a strong contribution during that run, scoring twice and providing two assists.
Hamburg confirmed that Gronbaek will now face a spell on the sidelines as he begins his recovery from the injury.