Without help, he would have most likely failed the exam.
He was definitely helped by Prof. Van der Werghe and Dean Fogg, but most likely,
Jane helped him by getting him into Brakebills and holding Julia back.
Other spoilers will be unmarked!
Jane helped him get to Brakebills
Under the guise of the Paramedic, Jane gave Quentin the manila envelope which lead him to Brakebills:
Unlike practically everybody, she seemed more interested in him than in Jomes. "You know, I think this guy might have left something for you."
She picked up two manila envelopes, document-size, off a marble-topped side table. Quentin frowned.
(The Magicians, "Brooklyn")
Quentin's envelope contains a book which contains a note. That note has no other purpose than bringing him to Brakebills, as we learn later in the book, so without it, Quentin would probably not have gotten to Brakebills:
There was a note tucked into the pages, the same one the wind has snatched away from him the first time he tried to read it. But all it said was:
(The Magicians, "The Retreat")
The Dean doesn't know exactly why he's there
"Oh, we have a device for that, a globe." [...] "It finds young people like yourself who have an aptitude for magic - essentially it senses magic being performed, often inadvertently, by unregistered sorceress, of which you are one. I suppose it must have picked up that Wandering Nickel trick of yours."
(The Magicians, "Eliot")
No one questions why Quentin arrived at Brakebills, but it seems that no one knows exactly why, either. It's not unlikely that had he not taken the book and note, he wouldn't even have gotten an invitation to Brakebills at all.
He only passed the practical test because the Dean was desperate to find the 20th student
After the practical test, no one talks to Quentin, but he overhears this:
"If we don't have a quorum, we'll simply send them all home and skip a year." Fogg's genteel reserve was decaying. [...] "There is a Twentieth, Melanie. We go through this every year, and we will empty every high school and middle school and juvenile detention center on the continent till we find him or her or it."[...]
The door opened a crack, and for an instant a worried face peered at him - it was Quentin's first examiner, the dark-haired European lady with the clever fingers. He opened his mouth to ask about a phone - his cell was down to one useless flickering bar - but the door shut again.
(The Magicians, "Brakebills")
At this point, Quentin has already failed. Melanie, the first examiner, didn't tell him, because she was the only one who has seen a hint of magic during his practical examination.
Additionally, the Dean was desperate to find a 20th student, a number which is somehow important:
It was an immutable fact of life at Brakebills that there were always twenty students per class, no more and no fewer.
(The Magicians, "Snow")
It's also obvious that what happens next is not part of the normal exam:
"-a Candidate? Fine," he said sarcastically. "Let's see a Candidate. And bring some Goddamn candles!" He sat down at the table. His shirt was translucent with sweat. It was not impossible that he'd had a drink between now and the last time Quentin had seen him.
(The Magicians, "Brakebills")
Fogg almost does the magic for Quentin:
"Stop fucking with us, Quentin!" Fogg barked. He snapped his fingers. "Come on. Wake up!"
He reached across the table and grabbed Quentin's hands roughly. [...] He was moving Quentin's fingers, physically forcing them into positions they didn't want to be in.
"Like this," he was saying. "Like this. Like this."
(The Magicians, "Brakebills")
Then, finally, Quentin performs some magic and is admitted.
Even after his admission, Quentin was lagging behind
There is more evidence that Quentin should have failed the exam. The first weeks describe him as one of the weakest students in his year. For example here:
Quentin stayed until his voice was hoarse and his fingers were on fire, until the light in the windows had softened and changed color and then sunk away completely, until his empty stomach ached, and dinner had been served and cleared away in the distant dining room. He stayed until his face was warm with shame, and all but four other people had stood up - some of them pumped their fists in the air and said yesssss!!- and left the classroom. [...] Finally Quentin said the chant and made the motions [...] and was rewarded by the sight of his marble wobbling, very slightly but unmistakably, of its own volition.
(The Magicians, "Magic")
Bonus Points: Julia should have passed her Exam
We know that Julia was there for the exam, but we do not learn when exactly she failed and why. This is her opinion on the matter:
"I was supposed to get in. I know I was. It was a mistake. Believe me, it was." Her large eyes tried to burn into his. "I'm like you. I can do real magic. See? That's why they couldn't make me forget!"
(The Magicians, "Alice")
While she is obviously prejudiced, she does have a point. Why did the memory spell not work on her? But more importantly:
From Fogg's quote above we know that each year 20 people are "found", no more, no less. But somehow, Julia, being very talented with magic (something we learn in the later books) botched the exam. We don't get any more evidence than this, but I think that it's clear that Julia should have been the 20th, and Quentin should not have been there at all.
Bonus Points: To pull this off, Jane probably only manipulated a single person
It was Melanie Van der Weghe who discovered Quentin's magic abilities. We don't know a lot about the practical examinations, but she was very patient with him, giving him a lot of time to perform his card tricks before she finally detects his magic abilities.
It was also her who suggested that Quentin moved up to Second Year together with Penny and Alice, something that others also see as an indication of Jane's influence.
She also made sure Alice got a chance to be interviewed, even though she wasn't Invited, and Fogg was against it.
Penny also gets personal sessions with her. He is able to get to the City which connects them to Fillory, and it is not a stretch to believe that Melanie told him how to come there, even though he claims that it's the other way round:
"I always thought it would be Melanie who came here the first time, but she could never quite work the spells. I tried to teach her, but she's not quite strong enough. Almost, though."
(The Magicians, "The Neither Lands")
Melanie Van der Werghe is described as having "a very slight European accent". We do not have more evidence to go on, but this might be an indication that she knew Jane, or might have been impersonated by her.