"quickly maturing" is correct and usable in written English.
You can use it when referring to something that is changing or developing at a rapid pace. For example, "The quickly maturing tech industry is seeing new innovations every day."
The field of glass science is quickly maturing from a purely empirical science to one built upon rigorous fundamental physics.
Or maybe Raef LaFrentz is quickly maturing into an All-Star and Avery Johnson still has enough left in his legs.
As an independent, self-reliant teenage girl, Naomi provides a model for any quickly maturing questioner, even those who don't share her questions.
Given the importance of as-built documentation, the quickly maturing technology of building information modeling (BIM), an object-oriented information-integration platform, has led to the emergence of concurrent as-built documentation during the construction phase.
Web and grid services are quickly maturing as a technology that allows for the integration of applications belonging to different administrative domains, enabling much faster and more efficient business-to-business arrangements.
And in a few years' time the length of time it might take to digest Friends Life opportunities abroad may well have dried up and quickly maturing markets could be hard to re-enter.
In this stimulating environment Hedberg quickly matured, increasingly abandoning traditional modelling techniques in favour of casting from plaster moulds.
Debuting in 1990 as the MIT $10K Entrepreneurship Competition, it quickly matured into one of the largest and most well-known business plan competitions in the world.
And they quickly matured into a self-contained ecosystem, with their own species – mall rats, mall cops, mall walkers – and an annual feeding frenzy known as Black Friday.
Their sound quickly matured, taking R&B next-level with 1994's sophisticated urban pop groundbreaker CrazySexyCool – the first girl-group album in music history to reach diamond status.
Unlike most muscles, heart tissue does not regenerate after it is damaged, because it contains no reserve of embryonic muscle cells, or myoblasts, that could quickly mature and restore the heart's normal function.
