I suppose Dumbledore (or someone else) had to state out the cause of Quirrell's death to the Ministry. Since no one has publicly claimed Voldemort has returned until the death of Cedric Diggory three years later, what did he tell them?
Probably nothing (at least officially).
Dumbledore probably tried to keep Quirrell's death as quiet as possible, at least outside the confines of Hogwarts. He couldn't keep the news completely contained, though. The Hogwarts gossip train had seen to that.
"What happened down in the dungeons between you and Professor Quirrell is a complete secret, so, naturally, the whole school knows."
(Philosopher's Stone, Chapter 17, The Man with Two Faces).
Although the full details might not have made it into the rumours...
It was one of those rare occasions when the true story is even more strange and exciting than the wild rumours. Harry told them everything: Quirrell; the Mirror; the Stone and Voldemort. Ron and Hermione were a very good audience; they gasped in all the right places and, when Harry told them what was under Quirrell's turban, Hermione screamed out loud.
(Philosopher's Stone, Chapter 17, The Man with Two Faces).
It seems that only Harry, Ron and Hermione (and probably the teaching staff) know that Quirrell is actually dead. Dumbledore certainly doesn't make a public announcement or lead any kind of public mourning in front of the whole school in the way he did for Cedric Diggory. That's probably because of the kind of man Quirrell was, and who he sided with. Instead, the mood of the feast was largely celebratory because of Gryffindor ending Slytherin's domination of the House Cup.
Quirrell's absence was probably only noticed by a few eagle-eyed students, who may have assumed that Quirrell had resigned or left before the end-of-year feast for some reason. Most students and parents may have only realised that there was a vacancy once Gilderoy Lockhart was announced as the new teacher (although they may have become accustomed to regular staff changes due to the DADA curse).
So the students may have known about Quirrell's death from the rumours but chances are most students didn't know about what happened to Quirrell in any great detail, or even necessarily notice his absence.
What did Dumbledore tell the Ministry?
We know that Dumbledore and Fudge were in regular contact, at least earlier on in Fudge's career.
"Deep down, Fudge knows Dumbledore's much cleverer than he is, a much more powerful wizard, and in the early days of his Ministry he was forever asking Dumbledore for help and advice," said Lupin.
(Order of the Phoenix, Chapter 5, The Order of the Phoenix).
However, these letters seem to have mostly been about the Ministry, not Hogwarts. Fudge says that he's always given Dumbledore "free reign", implying perhaps that he turned a blind eye to the day-to-day running of Hogwarts.
"Now, see here, Dumbledore," he said, waving a threatening finger. "I've given you free reign, always. I've had a lot of respect for you. I might not have agreed with some of your decisions, but I've kept quiet. There aren't many who'd have let you hire werewolves, or keep Hagrid, or decide what to teach your students, without reference to the Ministry."
(Goblet of Fire, Chapter 36, The Parting of the Ways).
So I think the chances are that Dumbledore wouldn't have wanted to tell the Ministry about the whole affair with Quirrell and the Stone, and that Fudge wasn't paying enough attention to notice it. Voldemort was thwarted so Dumbledore probably didn't think that it was worth alarming the whole wizarding world over a close shave. The Stone was destroyed so there was nothing that the Ministry could really do about the situation. And telling the Ministry wouldn't have reflected well upon the school's reputation and would've involved a lot of awkward questions. Why was a teacher who was possessed by Voldemort allowed to teach children? Why didn't Dumbledore do something sooner, after the Halloween troll incident and the attack on Harry during the Quidditch game? Why was Hogwarts allowed to serve as a hiding place for the Philosopher's Stone when Dark wizards were known to be after it? Why were three eleven year-olds down in the dungeons unsupervised and in mortal peril? And so on and so on. I think that Dumbledore would have much preferred to hush the whole thing up.
Remember that the Ministry didn't really interfere with staff appointments and the like until Order of the Phoenix. It's doubtful that the Ministry's awareness of the situation would have gone much beyond 'Oh look, they've hired a new Defence Against the Dark Arts teacher'.
We know from the conviction of Morfin that the Ministry does investigate suspicious deaths. They would have liked to investigate Quirrell's death, I'm sure, if they'd had the chance. The fact that we don't hear about any such investigation I think suggests that they didn't know that Quirrell had died.
We do see the Ministry interfering at Hogwarts in the case of Hagrid. However, this was only to temporarily arrest Hagrid on the off-chance that he was somehow involved in the attacks. It was largely done to curtail public political pressure and to satisfy the school governors.
"Look, Albus," said Fudge, uncomfortably. "Hagrid's record is against him. Ministry's got to do something - the school governors have been in touch."
(Chamber of Secrets, Chapter 14, Cornelius Fudge).
It seems that, unlike the Chamber of Secrets attacks, the Quirrell incident never got out to the school governors, the Ministry or the public at large. All that anyone had to go on were the unsubstantiated rumours of schoolchildren - and we know how infrequently adults in authority value the opinions of children in these books.
Probably just "Mr. Quirrell died trying to steal the Philisopher's Stone". Voldemort did not return, so there was no need to convince anyone that he had. Quirrell could've tried to steal the stone without being possesed by some evil master, eternal life might be tempting for most people.
Since no pupils were in danger and Dumbledore could not be held responsible for criminal activities of his staff, I don't think this would be a reason for the ministry to perform investigations about anything but the attempted theft.