The phrase "after quoting" is correct and usable in written English.
You can use it when you want to emphasize the fact that something has already been quoted or referenced previously. For example: "After quoting the author, I discussed the key points of the article."
After quoting extensively from his speeches and writings in the opening session, Mr. Blodgett cued up a video to let the man himself explain.
As he says, after quoting from a 1969 article, Even then people were screaming that no one would participate.
After quoting this passage from my Comment — — Chait remarks, "Hertzberg doesn't really explain this assertion and I don't understand it".
"There is no middle ground," Billy's son, Franklin Graham, told me in Khartoum, in 2003, after quoting that passage.
After quoting wire service reports that the Foreign Ministry had been engulfed by demonstrators, it followed up with denials from Egyptian officials.
He also came in for criticism after quoting Hitler's Mein Kampf at a hustings in order to make a point about the EU being undemocratic.
After quoting telegrams he'd received from the White House, he thanked those who had supported him, acknowledged the hard road ahead and committed himself to the job.
Soon after quoting this, Haffenden comments: "Throughout his adult career Empson maintained consistently anti-censorious views: every issue, he believed, should be available for literary treatment and criticism".
After quoting Rommel, Walsh advises the no-huddle as the way to handle "extreme" defensive tactics, including "an extensive use of high-risk blitzing".
After quoting the retired general Anthony Zinni on his deep admiration for Powell, for instance, she allows him to add: "Powell is a pretty ambitious guy.
