I really don't know why you would want to control fluorescent lights with a dimmer controlling a relay. It is a convoluted way of replacing a single pole switch.
The above is what you are asking to do.
A dimmer and fluorescent lights do no work together very well unless you have dimmable ballasts. Adding a relay in the mix does not solve anything.
The link provided is to a relay that is rated for controlling ballasts totaling up to 450VA. I don't know the ratings of your ballasts to even begin to say if it's compatible. The relay is NOT rated for electronic ballasts. The relay will accept either (10-30 volts either AC or DC) or (120V AC) to control the switching of the relay and the relay has a normally open and a normally closed set of contacts to supply your load.
If you don't understand relays, the basic principle is to use one source of power to switch another source of power. Typically used as 24V source switching 120V or 240V loads. Or also typically used as 24V or 120V source switching 480V loads.