I'm developing a telephone call recorder application, does this modem detect when the remote telephone goes on-hook and off-hook?

The modem is not capable of detecting when the remote side has gone Off-Hook, or On-Hook.

Silence detection may help you detect when the remote side answers the phone. Silence can be detected in Voice Receive mode, but not in Speakerphone mode. There are a couple different commands that can be experimented with to adjust when the 'OK' response is sent by the MT5656ZDXV in originating mode, indicating that the modem is ready to receive and transmit voice. These commands are timers, based on the Ringback cadence, and when they expire, it is assumed that the remote side has answered. The commands are +VRA, and +VRN. With +VRN, the timer starts after dialing, and if a Ringback is not detected before the timer expires, then the modem responds 'OK'. The fact that another ringback was not detected assumes the remote end answered. If a Ringback is detected, then the timer resets. The +VRA command works pretty much the same way, except that the timer doesn't start until after the first Ringback is detected.

These commands are explained in the AT Command Reference Guide at:

http://www.multitech.com/en_US/DOCUMENTS/Families/MultiModemZDX/manuals

These commands may aid you in determining when the remote side has answered (gone Off-Hook), but there isn't any way to determine when the remote end has gone back On-Hook.