My children receive WIC and Quest benefits from the state. Does that mean they receive public assistance?

Yes. Those are government programs.

Do we qualify for EIC? Or do I omit them?

To qualify for and claim the Earned Income Credit you must:

• Have earned income; and
• Have been a U.S. citizen or resident alien for the entire tax year; and
• Have a valid Social Security number (not an ITIN) for yourself, your spouse (if filing jointly), and any qualifying children on your return; and
• Not have investment income exceeding $3,600; and
• Not be filing a Form 2555 or 2555-EZ; and
• File a return with the Single, Married Filing Jointly, Head of Household, or Qualifying Widower filing status, even if you're not required to file a return

Thank you. What about for my dependents.? 

To see if your dependents make you eligible for EIC with a qualifying child, please see the TurboTax Help article “What is a qualifying child for the Earned Income Credit (EIC) in 2019?” for more information.