There are no ways to tell that zebra finch is going to lay egg except when it start nesting. Zebra finch can lay 3-6 eggs and then starts hatching which can be of 12-15 days. Eggs are hatched by both male and female.

 Well being a breeder myself , I have had 2 white zebra finches (tito & twinkee) who have had 9 babies. And there ARE ways to tell if the female is going to lay egg(s).
But it depends.. Is it one female bird, or a female and a male together? Because a female zebra finch will lay eggs whether a mate is present or not.
IF THERE IS ONLY 1 FINCH:
You can tell if shes going to lay eggs, by her size. You will notice that your bird will get very fatter, and start eating more.
 
IF THERE IS A MALE AND FEMALE PRESENT:
1st you will see them mate.
If you dont then its ok, but either way you may notice that the male and sometimes the female will start trying to make a nest out of any materials they can find. Once again the female will get fatter and she will be eating more.
 
She should (about a week or so later) lay one egg per day, with a max of 12 eggs in one clutch.