Do I need to pay quarterly taxes both to the state and federal government, or just the state, if that's what appeared on my return this year?

[Vanessa]

If you do not have enough withheld from your wages or other income, then yes, you will need to pay quarterly taxes to both state and federal.

If you did not owe anything to federal but did to the state and your situation and withholdings are not changing for 2019, then you would only need to pay quarterly taxes to your state. 

Basically, both require you to make quarterly payments if you are going to owe them.