Update: I figured out a workaround for the time being and wanted to post on the off chance somebody has the same issue.
I started adding watches to inspect the results of the various methods of the INote interface when a balloon was selected. I discovered that a balloon with the "Quantity" checkbox shown returns a value of 3 when INote.GetTextCount is called. When I call INote.GetTextAtIndex(3), it returns a concatenated string containing the quantity (or overridden value) and the Denotation string.
I haven't found a case where this information would not be stored at index 3, but it seems like a sort of hokey way of accessing that data.
Here is a simple macro I wrote to get the selected balloon so I could interrogate the object. I just set a breakpoint on the End Sub line.
Dim swApp As Object
Dim swDocument As IModelDoc2
Dim selMgr As ISelectionMgr
Dim balloons As Object
Dim balloon As INote
Dim bStatus As Boolean
Set swApp = Application.SldWorks
Set swDocument = swApp.ActiveDoc
Set selMgr = swDocument.SelectionManager
Set balloon = selMgr.GetSelectedObject6(1, -1)
Set myNote = Part.Extension.EditBalloonProperties(swBalloonStyle_e.swBS_Circular, swBalloonFit_e.swBF_2Chars, swBalloonTextContent_e.swBalloonTextCustom, "2.1", 0, "", 0.01016, False, 1, "X")