My 3 children are American citizens, I am Hungarian but hold a Green card. We live in Hungary for most of the year (11 months). My only income is the child support and alimony that comes to my Hungarian account from my ex-husband. I don't have a Stateside address. Do I have to file any tax paperwork in the States?

.If your total worldwide income exceeds the filing requirement, you must file your U.S. tax return. On your return, you must report your world wide income. If you paid taxes in the foreign country, you will claim credit for taxes paid in the foreign country by filing Form 1116 or can file form 2555 Foreign Earned Income Exclusion..
Read: Your U.S. Tax Return: U.S. Citizen or Resident with Foreign Income

Note that, while alimony is taxable, child support is NOT!