AT$ICCID? Should do the trick I think. You were close :)
UPDATE #1:
Ok, on further looking around it seems this modem doesn't even have anything implemented to get the ICCID. Based upon http://www.farnell.com/datasheets/876256.pdf which is in theory the AT Command Set reference for the device.
UPDATE #2:
Based on the suggested answer from hlovdal here is the command you could use:
AT+CRSM=176,12258,0,0,10
Tried this just now on a Sierra Wireless modem with success! <176> is the command type which is READ_BINARY, <12258> is the file location of the ICCID on the SIM card and the rest is parameters and response code I think.
You will get a response back something like:
+CRSM: 144,0,"2143658709"
The "2143658709" is in binary coded decimal format with switched digits. So in this example it would turn out to be "1234567890".
Learning some new cool stuff here :)