This can usually be done in a modem chat script, but exactly what command to use varies modem-by-modem. Also, it's been so long I can't remember more about how it's done. Hope this helps anyway.
Come to think of it, there's bound to be a ms-windows or DOS guide for silencing this modem. Putting roughly the same commands in the chat script should work. Won't solve the problem with alsamixer, tho.