ice cube relay help wiring and will it work

If the HOA switch outputs 24V DC, and the irrigation controller needs a contact-closure in a 24V AC circuit to operate, then any normal 24V DC relay will work. I.e. the relay must have a 24V DC coil, as AC and DC coils are different. The fact that it is switching an AC circuit to the irrigation controller instead of a DC circuit doesn't matter, as its contacts will be equally suitable for both.

I don't know what the output of the HOA switch is intended to drive, but if it is only a low-current output of a fraction of an amp, it might be worth getting a relay with a built-in back-EMF protection diode or adding one yourself, to ensure that the output is not damaged by the back-EMF from the relay coil when it switches off. If the output is intended to drive motors, pumps etc then it won't be a problem.

Also, I don't know what the control input to the irrigation controller consists of. If it is just a logic / control circuit that operates the solenoid valves / pumps by its own internal relays, then there will be very little current and the contact rating of your relay won't be important. But if for any reason that input is switching motors etc directly, then you would need to make sure your relay contacts have adquate current rating.

This might be easier to advise if you post the model numbers of the equipment.