Generally not. You can discharge debts but not payments imposed on you that are essentially punishment for some behavior or circumstance. Remember the courts are part of the government and the government always gets paid if there's anything there of value.
While it is true that some court ordered payments are non dischargable (specifically for crimes), it is NOT true that all debts to the government are nondischargeable. For example, if you owed money to governmental agency due to a loan or overpayment of benefits (not including social security or student loans), those would be dischargeable like a normal loan. Restitution is not dischargeable. Criminal Judgments are not dischargeable. This is not quite the same as owing the government money for something that has nothing to do with court.
It depends on what kind of court fines. Generally, under Chapter 7, fines of governmental units are not dischargable. Neither is criminal restitution. Under Chapter 13, fines may be repaid, but generally, punitive action by such agencies as criminal courts for failure to pay court-ordered fines are not stayed.
Parking tickets and tollway fines for not paying tolls are usually dischargable under Chapter 13, but only if you provide for them in your Chapter 13 debt repayment plan, and complete the plan and receive a discharge.
This is general advice only and does not create an attorney client relationship. You should consult a local attorney for advice on your specific situation.