Translation:The children lie down in the bed and chat.
In that case: "into the bed". Because the Hungarian sentence is talking about them getting into bed, not already being there.
British English prefers "get into bed' and "lying in bed" without the definite article unless you want to stress a specific bed. There are a lot of set expressions indicating motion that dispense with the definite article and sound very odd with them.
Go to work/school/hospital/bed etc
What time did you get to work/school/bed etc?