"were quick to realize" is correct and is used in written English.
You can use it when someone acted quickly or quickly understood something. Example: My friends were quick to realize that I was in need of help.
They were quick to realize their real affinity was with the Pres.
"From the Games in Los Angeles onward, the Chinese were quick to realize that, if you want to win at the Olympics, you have to use drugs".
The queen withstanding, royals and cafe society were quick to realize that when working with Spry, minimal interference netted maximal results.
Dealers were quick to realize that they could repackage this special sage and legally sell it as a "medicinal herbal supplement" through a loophole in the new law.
As China-watchers were quick to realize, President Obama did not even once mention the "New Type of Great Power Relations" on his recent trip to Beijing.
Of course, companies and entrepreneurs alike were quick to realize that whoever could get their websites to show up for relevant queries on the 1st page of Google would generate significant additional revenue.
But what we saw during Trump's so-called press conference is that for all this vaunted nimbleness, the editorial judgment of cable news producers lags well behind their counterparts outside the studio, who were quicker to realize they were being conned.
Physicists, who among other things invented the World Wide Web, have been quick to realize that their lexicon is no longer too obscure for pop songs.
He was known as an intimidating boss, whose tirades sometimes reduced his underlings to tears, but, unlike some of his colleagues, he had been quick to realize the threat that Spitzer's investigation represented.
While Warburg, like so many other wealthy German Jews, misjudged the threat posed by the rise of the National Socialists, once Hitler came to power he was quick to realize how dangerous the situation had become, and in 1934 he emigrated.
Pawson was quick to realize that I am very keen to understand many biological problems.
