Father wants custody of child and he told judge that he has his own place but the fact is he doesn't have his own place and lives with girlfriend whom he constantly fights with her. Can that be presented in court against him?

One has to define "own place" if he and his girlfriend live together and have an apartment, that may or may not be considered his own place.

But in a custody hearing, you can challenge his statements, ask him to provide proof, in forms of lease, or rental statements.

But merely living with his girl friend does not mean he could not get at least joint custody, or custody ( if there is a reason the other parent is not a fit parent)

And the fact they "fight" if you mean argue,not phsycal fight, if they "fight" are there police reports of this to use as evidence

Ask an attorney. Basically I believe as long as there is a bedroom for the child and the child is treated appropriately (fed, sheltered, not beaten, not around intoxicated parents, not put in harm's way) there is no reason for witholding visitation rights.
Honestly, regardless of the outcome, you may want to go to mediation before you ever even see a judge. It may be better for the kid if everything is worked out before an issue arises and then, each parent knows what is expected of them.