Even we can refresh particular connection and in turn it will refresh all the pivots linked to it.
For this code I have created slicer from table present in Excel:
This code is for Slicer from DB:
Sub UpdateConnection()
    Dim ServerName As String
    Dim ConnectionString As String
    Dim DatabaseNameCount As Integer
    DatabaseNameCount = ActiveWorkbook.SlicerCaches("Slicer_Name").VisibleSlicerItems.Count
    If DatabaseNameCount = 1 Then
        ServerName = ActiveWorkbook.SlicerCaches("Slicer_Name").VisibleSlicerItems.Item(1).Name
        ConnectionString = GetConnectionString(ServerName)
        UpdateAllQueryTableConnections ConnectionString
    Else
        MsgBox "Please Select One Value", vbOKOnly, "Slicer Info"
    End If
End Sub
This code is for Slicer created from Excel table present in same workbook:
Sub UpdateConnection()
        Dim ServerName As String
        Dim ServerNameRaw As String
        Dim CubeName As String
        Dim CubeNameRaw As String
        Dim ConnectionString As String
        ServerNameRaw = ActiveWorkbook.SlicerCaches("Slicer_ServerName").VisibleSlicerItemsList(1)
        ServerName = Replace(Split(ServerNameRaw, "[")(3), "]", "")
        CubeNameRaw = ActiveWorkbook.SlicerCaches("Slicer_CubeName").VisibleSlicerItemsList(1)
        CubeName = Replace(Split(CubeNameRaw, "[")(3), "]", "")
        If CubeName = "All" Or ServerName = "All" Then
            MsgBox "Please Select One Cube and Server Name", vbOKOnly, "Slicer Info"
        Else
            ConnectionString = GetConnectionString(ServerName, CubeName)
            UpdateAllQueryTableConnections ConnectionString, CubeName
        End If
    End Sub
Common code to create connection and update connection for desired Initial Catalog:
Function GetConnectionString(ServerName As String, CubeName As String)
        Dim result As String
        result = "OLEDB;Provider=MSOLAP.5;Integrated Security=SSPI;Persist Security Info=True;Initial Catalog=" & CubeName & ";Data Source=" & ServerName & ";MDX Compatibility=1;Safety Options=2;MDX Missing Member Mode=Error;Update Isolation Level=2"
        '"OLEDB;Provider=SQLOLEDB.1;Integrated Security=SSPI;Persist Security Info=True;Initial Catalog=" & CubeName & ";Data Source=" & ServerName & ";Use Procedure for Prepare=1;Auto Translate=True;Packet Size=4096;Use Encryption for Data=False;Tag with column collation when possible=False"
        GetConnectionString = result
    End Function
    Sub UpdateAllQueryTableConnections(ConnectionString As String, CubeName As String)
        Dim cn As WorkbookConnection
        Dim oledbCn As OLEDBConnection
        Dim Count As Integer, i As Integer
        Dim DBName As String
        DBName = "Initial Catalog=" + CubeName
        Count = 0
        For Each cn In ThisWorkbook.Connections
            If cn.Name = "ThisWorkbookDataModel" Then
                Exit For
            End If
            oTmp = Split(cn.OLEDBConnection.Connection, ";")
            For i = 0 To UBound(oTmp) - 1
                If InStr(1, oTmp(i), DBName, vbTextCompare) = 1 Then
                    Set oledbCn = cn.OLEDBConnection
                    oledbCn.SavePassword = True
                    oledbCn.Connection = ConnectionString
                    Count = Count + 1
                End If
            Next
        Next
        If Count = 0 Then
             MsgBox "Nothing to update", vbOKOnly, "Update Connection"
        ElseIf Count > 0 Then
            MsgBox "Connection Updated Successfully", vbOKOnly, "Update Connection"
        End If
    End Sub