You can either have the modem pass the message directly and
immediately to the serial port (rather than store it
internally), or simply have the modem send an unsolicited
notification to the serial port.

There are quite a few options as well as device-specific
implementation uncertainties however, so I can't suggest a set
of exact commands guaranteed to do just what you want.

GSM specification 07.05 contains details of the SMS-related
"AT+C" commands, but your modem documentation should have actual
 supported syntax and parameter lists.

You'll need to decide between text-mode and PDU-mode.  See
"AT+CMGF".

Use "AT+CNMI" to route either the messages themselves, or simply
SMS notifications, to the serial port as either "+CMT: " or
"+CMTI: "  unsolicited result codes respectively.  You may need
to use the "AT+CNMA" command in combination with this.

This can be complex stuff for the uninitiated, so don't be shy
about asking further questions if you get stuck.

John