Here's another solution that uses some of VBA's built in date functions and stores all the date data in an array for comparison, which may give better performance if you get a lot of data:
Public Sub MoveData(MonthNum As Integer, FromSheet As Worksheet, ToSheet As Worksheet)
Const DateCol = "A" 'column where dates are store
Const DestCol = "A" 'destination column where dates are stored. We use this column to find the last populated row in ToSheet
Const FirstRow = 2 'first row where date data is stored
'Copy range of values to Dates array
Dates = FromSheet.Range(DateCol & CStr(FirstRow) & ":" & DateCol & CStr(FromSheet.Range(DateCol & CStr(FromSheet.Rows.Count)).End(xlUp).Row)).Value
Dim i As Integer
For i = LBound(Dates) To UBound(Dates)
If IsDate(Dates(i, 1)) Then
If Month(CDate(Dates(i, 1))) = MonthNum Then
Dim CurrRow As Long
'get the current row number in the worksheet
CurrRow = FirstRow + i - 1
Dim DestRow As Long
'get the destination row
DestRow = ToSheet.Range(DestCol & CStr(ToSheet.Rows.Count)).End(xlUp).Row + 1
'copy row CurrRow in FromSheet to row DestRow in ToSheet
FromSheet.Range(CStr(CurrRow) & ":" & CStr(CurrRow)).Copy ToSheet.Range(DestCol & CStr(DestRow))
End If
End If
Next i
End Sub