what about "they age quick"
In standard English, quick/quicker/quickest are adjectives, and the corresponding adverbs are quickly/more quickly/most quickly.
Some years ago, to use the adjective in place of the adverb was considered a hallmark of illiterate, uneducated speech. Nowadays, however, standards have changed. Or is it usage that has changed?
My answer -- They are getting old quickly -- was accepted, so I guess this issue has been resolved. Yay!
Is there a reason that "They are growing up fast" is not accepted?
I'd say that "growing up" is not really right. The DL sentence could be about cheeses or whiskey aging in a cellar. It is not necessarily about human beings or animals "growing up".
Even restricted to human beings, the DL sentence could be about adult, fully-grown humans getting even older, rather than about children maturing.