The answer is most likely "Yes"
You already know that
the Paramedic is Jane Chatwin, and that Quentin is not the first she is trying to put in the way of The Beast.
This is the part where Quentin and Fogg discuss the paramedic:
"What about the woman I met, with the braids. The paramedic. Was she a scout, too?
Fogg frowned. "With braids? You saw her?
"Well, yes. Right before I came here. Didn't you send her?"
Fogg's face became studiously empty.
"In a manner of speaking. She's a special case. Works on an independent basis. Freelance, you might say."
(The Magicians, "Eliot")
Fogg easily identifies her even though Quentin has been given a very lousy description ("Woman with braids, paramedic, met her once"). Since Fogg was actually surprised that the paramedic was there, we know that Fogg doesn't know all the circumstances that lead Quentin to Brakebills.
Fogg knew who she was, and her habit of selecting people for
the fight against the beast
is what he refers to as "working on an independent basis". He probably wasn't happy with it, and kept the emotions out of his face as they talked.
So how do we know that he knows exactly who she is? Well, she hides his ticket to Brakebills in a book:
The first page, handwritten in ink, read:
Book Six of Fillory and Further
The ink had gone brown with age. The Magicians was not the name of any book by Christopher Plover that Quentin knew of. And any good nerd knew that there were only five books in the Fillory series.
When he turned the page a piece of white notepaper, folded over once, flew out and slipped away on the wind.
(The Magicians, "Brooklyn")
We learn later that the book
Was written by Jane herself, and contains information about Martin's transformation into the Beast.
So, the paramedic tries to make sure that Quentin goes to brakebills and
gets information about the enemy he is supposed to face.
Knowing Quentin, he would have absorbed the contents of the book within days after getting it. However, the book is gone after Quentin decides to join Brakebills (and also after Fogg learns that the paramedic was involved):
The only thing missing was the book, the notebook the paramedic gave him. That was nowhere to be found. He'd left it in the exam room on the assumption that he'd be going back there, but when he finally did it was gone. Dean Fogg and the butler pleaded ignorance.
(The Magicians, "Eliot")
The most likely explanation is that Fogg knew who she was and took the book away, hoping to
spare Quentin the fight against her brother.
We do see the book back again later in the book, when he
discusses time travelling with Jane Chatwin. On that book, the note, which has flown away and guided Quentin to Brakebills is inexplicably back tucked between the pages, which is an indication that this book is either from an alternate timeline, or it's the one that travels back in time with Jane and is then handed to Quentin.