The voltage on a flash contact pair is very high, so normally a component called an SCR is used to drive it. This can be done with this sort of circuit:-
That uses a photo transistor to act as a slave flash. Just remove that and feed an Arduino output signal into the 1K resistor. Don't forget to connect the 0V to the GND of the Arduino.
Use a meter on your flash contacts to see which is the ground and which the hot end.
I did search for a solution but wasn't able to find anything.
I searched for :-
photo flash trigger schematic
and came up with lots of stuff.