"quickly flee" is correct and usable in written English.
You can use it when you want to express the urgency of a person or group of people fleeing from a dangerous or undesirable situation. Example: When the fire alarm went off, the students quickly fled the building.
Some security sources speculate that the abductors killed them so that they could quickly flee, after the pickup truck broke down only 12 kilometers from Kidal.
And he recounts how white Zimbabweans and many middle-class blacks readied "gap bags" in case they needed to quickly flee.
Khyber borders Peshawar, where the school massacre happened, and militants have traditionally attacked the city only to quickly flee into the tribal region where police cannot chase them.
Then there is the capricious nature of the filmmaking business, which has become increasingly mobile and will quickly flee to the national or local government offering the best incentives.
Once allegations surfaced in 2008, Leifer was stood down by a committee linked to the school and plane tickets were purchased so she could quickly flee the country, the court heard.
With Morton's aid, the Republicans fled to Madison where they could quickly flee into Kentucky should the Democrats attempt to forcibly return them to the capitol.
Opportunity quickly flees.
Jennifer, his beautiful and strong-willed new wife, quickly fled, taking her baby with her.
The mice froze or quickly fled to dark corners, behaviors considered by some researchers to signify anxiety.
Rather than face the music, Richardson admitted his guilt and quickly fled to Spain, never to return.
After hearing about the cease-fire, his mother and three sisters had decided to return home but quickly fled again, he said.
