Considering events happening in the first book, can Quirrell be considered a Death Eater?
Quirrell probably wouldn't be considered a Death Eater, just an ally.
Not all of the Dark Lord's allies are considered Death Eaters. Willingly supporting him doesn't seem to be the only criterion for being considered a Death Eater. To put it more simply, the Death Eaters were generally the ones who were in Lord Voldemort's official army, and were the ones who'd fight alongside him.
However, Quirrell found the Dark Lord while he was traveling, and doesn't seem to have been made an official Death Eater during that time. The Dark Lord was likely more concerned with his plan for resurrection than making Quirrell a Death Eater. He certainly wasn't a known supporter of the Dark Lord before that, because Dumbledore was willing to hire him as a teacher. In addition, it's questionable if Quirrell would have even wanted to become a Death Eater. It's unclear how willing Quirrell truly was to support the Dark Lord. The Dark Lord would probably not actually make someone he's been forcing to serve him from the beginning an official Death Eater. Even Peter Pettigrew, arguably the least willing and most cowardly of the Death Eaters, probably had to agree to become one at some point, even if he later regretted it.
This quote implies that he wasn't necessarily willing to serve the Dark Lord at first:
“He is with me wherever I go,’ said Quirrell quietly. ‘I met him when I travelled around the world. A foolish young man I was then, full of ridiculous ideas about good and evil. Lord Voldemort showed me how wrong I was. There is no good and evil, there is only power, and those too weak to seek it … Since then, I have served him faithfully, although I have let him down many times. He has had to be very hard on me.’ Quirrell shivered suddenly. ‘He does not forgive mistakes easily. When I failed to steal the Stone from Gringotts, he was most displeased. He punished me … decided he would have to keep a closer watch on me …”
However, the Dark Lord does refer to him as faithful, although that may be a natural effect of being possessed by the Dark Lord for such a long time, or possibly some form of mind control or weakening as well:
“See what I have become?’ the face said. ‘Mere shadow and vapour … I have form only when I can share another’s body … but there have always been those willing to let me into their hearts and minds … Unicorn blood has strengthened me, these past weeks … you saw faithful Quirrell drinking it for me in the Forest … and once I have the Elixir of Life, I will be able to create a body of my own … Now … why don’t you give me that Stone in your pocket?”
Narcissa supported him, up until she decided her son was more important than the Dark Lord, but she was never considered a Death Eater. However, she did help him in some ways, like letting Malfoy Manor be used as the Death Eaters' headquarters. She doesn't do as much as Quirrell, but she's still an ally of the Dark Lord at first.
There are other criteria as well, Death Eaters can be half-bloods or even mudbloods, but there are no non-humans who are known to have been Death Eaters. Fenrir Greyback was willing to support him, as were other werewolves, giants, and the Dementors, but none of them would be considered Death Eaters.
Quirrell was more of a tool, a vessel for the Dark Lord to use while regaining strength.
1) Voldemort was in a very weak state at this time, and cursing (or blessing, depending on your viewpoint) someone with a Dark Mark would probably require strength. Which he'd want to conserve.
2) Quirrell was hinted to be reluctant to follow through with what Voldemort was asking of him. He was a Ravenclaw, not a Slytherin. I think he was only in it for the knowledge, and got in over in his head. Or maybe he just cracked under the pressure. Either way, he didn't seem to be great Death Eater material, and Voldemort didn't need him to be one to control him. He was already living in the back of his head; a Dark Mark would be overkill.
He could be considered a Death Eater in spirit, but I don't think he actually was one.