In most states the county seat is where the courthouse complex is and that's where you'll end up going.
Here's the thing, UNLESS you're really convinced this baby is yours and you want to stay in the baby's life, you kind of have to wait for her to make the first strike (which would be filing for child support) and then you respond.
If you're really intent on being on the offensive and want to push for the paternity test then you will need to contact a lawyer, but expect to put out the fee for at least a few hours consultation with a lawyer, plus the cost of filing fees and the cost of the paternity test.
I would think (I'm NOT 100%, we really need a father whose gone through this in California OR a lawyer's input), that she has to bear the burden of the cost of the DNA test if it's court ordered and you aren't the father. You might have to split it with her if it's found to be your kid.
Basically, you're only course of action is to CUT OFF ALL CONTACT WITH HER, now, permanently. Force all communication to take place via email (so you have record of anything that she says to you). If she leaves you voice messages or texts, save anything that might be pertinent down the road.
Oh, and be absolutely civil to her. Do not do a Mel Gibson. Never have any contact with her in an agitated mental state.