The phrase "are quickly beginning" is correct and usable in written English.
It can be used to describe something that is starting to happen rapidly or is in the early stages of development.
Example: "The team are quickly beginning to see the results of their hard work as the project progresses."
Alternatives: "are rapidly starting" or "are swiftly commencing".
And they are quickly beginning to rival the candidates' own money operations in size and scope, setting off a fund-raising arms race that is changing the way presidential campaigns are financed and executed.
Eric Tholome, Director of Product Management for Google Developer, says that Enterprise is a key focus because companies are quickly beginning to adopt the web stack and cloud computing.
"Our local police are quickly beginning to resemble paramilitary forces.
The negatives are quickly beginning to outweigh the positives, so I'm sure it won't be long before I kick the cigarette habit.
Unfortunately, due to a Department of Defense DODD) Program that transfers surplus DOD equipment to state and local law enforcement, our local police are quickly beginning to resemble paramilitary forces".
Despite the enthusiasm, the fan rally was quickly beginning to take its toll.
In an attempt to save the doctor, a global letter-writing campaign was quickly begun, with pleas aimed at Gen. Pervez Musharraf, Pakistan's military ruler.
The wooded site was still uninhabited, but construction of a capitol building was quickly begun, and late in 1838 the territorial legislature held its first session in a nearby building.
A new, accurately detailed survey was quickly begun.
Sulla was elected dictator and, while Italy and all the provinces except Spain were quickly reduced, began a reign of terror (the "proscriptions"), in which hundreds of his enemies or those of his adherents were killed without trial, while their property went to enrich him and his friends.
Government experts, demographers and economists say that as Japan loses people -- and workers -- the only way for it to maintain its lofty living standards is to quickly begin accepting immigrants in far greater numbers and to abandon the age-old comfort of its near-uniform ethnicity.
