Fagnano, a transfer from Maine, was injured during a designed quarterback run in the second quarter of Saturday's 35-14 loss in Atlanta.
"It's a fairly significant injury. We didn't think it was that significant until they got a real good look at it with the MRI, so Friday he's scheduled for surgery and he's obviously pretty down about it," Mora said.
Ta'Quan Roberson, a transfer from Penn State, will take over the starting duties for UConn (0-2), which hosts FIU on Saturday.
Roberson completed 19 of 30 passes for 216 yards and two touchdowns in relief of Fagnano on Saturday night.
He returns to the starting role he lost after suffering a season-ending knee injury in UConn's opening game a season ago.
"Now this year, the guy ahead of Ta'Quan gets hurt and now Ta'Quan gets his chance again. We have a lot of confidence in him and we're really disappointed for Joe," Mora said.
Zion Turner, a sophomore who started 12 games for the Huskies last season, will be Roberson's backup.