On the cellular modems I have worked, the voice path was set up by default to interface to a microphone and speaker. However most had a separate PCM interface which allowed the cellular modem to be connected to a Bluetooth headset via a Bluetooth module, provided it had a compatible PCM interface.
According to this website, the format is typically a 64 kb/s log PCM format (A-law or μ-law) or a 64 kb/s CVSD (Continuous Variable Slope Delta Modulation). So it seems you should be able to digitally interface this with a computer, and decode the modulated audio.
On one of the chipsets I looked at (which is pretty typical), the PCM interface consists of 4 pins: PCM_SYNC, PCM_DOUT, PCM_DIN, and PCM_CLK. It supports linear (13-bit), A-Law (8-bit), and µ-Law (8-bit) companding algorithms. The AT commands to control the PCM interfaces all appear to be proprietary, unlike the standard (AT+) set of ITU-T V.250 commands supported across all GSM chipsets.
Besides using the PCM interface, the other way of course would be to feed the speaker output into an ADC of the micro. I have done the latter, when I needed to do echo cancelling on a speakerphone application (the cell module supposedly was going to have a function to do that, but it was not implemented in the firmware yet.)