24VAC to 12VDC momentary switch short
Looks like this board actually supplies power to the outputs, so you will need to connect something there to convert to a contact closure. 

A conservative suggestion would be to source a 240VAC-input interval timer module which would give you a timed contact closure (followed by a drop out) whenever power is applied. Just set the timer to a second or two to simulate a button press. 

You could probably just slap a 240VAC-input relay on the flashing output, but it's possible the controller would not like the inductive load and you'd have to attach a snubber or MOV.