Hello, My question is I have a trial coming up, I'm being charged with criminal damages to property 2nd. I have proof that I was at work. I over heard the assistant D.A telling my attorney that he don't want to try it and speak to someone in their office about it. The next day they called me and said we are going to trial. I have a public defender lawyer. Is it possible that they are trying to get more money by not getting it thrown out and going to trial?

No, first what type of public defender, most merely work for the county or state and are on salary, and have more cases then they can even think about. Most never want to go to trial.
Another type is an attorney that presents cases to that court, and they require them to take some cases free, ( no pay) so they really don't want extra work.

And the other are private attorneys paid a very small amount compared to their normal salary to do defense work

And in California, I understand some new attorneys do the work in exchange for insurance coverage

Most likely what the DA was saying he did not want to go to trial, so he wanted you to plead guilty to some lesser offense. That is what DA and public offenders normally do, if you are charged with that, they would move it to a lesser offense or a tresspass or something but you end up being guility of a crime.