Well, I finally got a better understanding of what triggers do on the FY3224S. There is no relationship to the counter even though all the trigger settings are accessed under the [Count] button on the front panel.
Setting a Trigger
A trigger is used to initiate a pulse of the CH1/MF waveform, n number of times, as defined by the CP_CNT (CyclePlus_Count) value. There are three triggering modes: manual (Manual Trigger), external (Ext Trigger), or channel CH2/SF frequency (CH2 Trigger). Here are the pertinent command that control triggering...
tt0 Trigger manually. A trigger event is initiated by physically depressing the [ADJ] knob on the front panel.
Note: A “manual” trigger event cannot be initiated programmatically.
tt1 Use the signal from the BNC connector labeled Input on the front panel for the trigger.
tt2 Use CH2/SF frequency as the trigger.
tnn Set number of waveform cycles to output (CP_CNT value), where n is the number of cycles.
Note: Changing the CP_CNT value programmatically causes the trigger mode to change to manual trigger mode.