you need to change your code to send a string, not a list
something like (untested and not necessarily correct from a syntax point of view)

Cmd.CommandText = "SP_Insert_WC000311A_PremFormData"

dim liststr as String
dim lists As List(Of String) = QuoteNums.ConvertAll(Of String)(Function(i As Integer) i.ToString())
dim liststr as String = String.Join(",", lists)
Cmd.Parameters.AddWithValue("@quotelist", liststr)

Cmd.Parameters.AddWithValue("@quotelist", QuoteNums.ToString.ToArray)