I was hired by a company in San Francisco early this year. They paid for all the moving expenses and I signed an agreement on reimbursing them proportionally if employment is terminated within two years.
The last few months, it has become very difficult to work with my manager. This person sent out emails making 'false accusations' and had generally poor business conduct (e.g. telling client the wrong thing, not having client's best interest in mind).
Can I quit and/or negotiate with them?

Hello.
If your manager engages in a conduct that is illegal, that can certainly be a leverage in your negotiation over quitting in exchange of release of your obligation to pay for moving expenses.

Thanks