(This got super long and I don't know how)
Alright, to sort of explain how the Qunari might react to mages attempting to convert, I'm going to set up a little analogy. Let's say, for the sake of argument, that being a mage in Thedas isn't magical for this analogy. Let's say that instead of being born with hands capable of magic, mages are born with, I dunno, maybe hands covered in Vaseline.
If you're having a hard time picturing that in your head, I've whipped together a quick concept to help you out.

I feel like maybe this image of Vaseline Man has made things a little more convoluted, but it's too late to go back on that.
Depending on where you're born and what race you are, your views on Vaseline Man up there are different. Most humans, for example, believe that it was Ancient Vaseline Man and his buddies who smeared their petroleum jelly-covered hands all over the Maker's Spiritual DisneyWorld, which is why he's suddenly stopped returning their calls. Tevinter humans believe this to an extent, too, but also that people born with the natural ability to secrete petroleum jelly are the master race. This analogy keeps getting more and more messed up.
In Qunari culture, being born with Vaseline hands immediately makes you a theat to every single other member of society, period. You're assigned a personal guard, an avaraad, almost ASAP to ensure you don't smear you delicate, greasy hands all over people and kill them violently in the process. It's the belief of everyone around you that if left unattended, even for a short time, that demons will be drawn to the softness of your little hands and will possess you. It's also believed that anyone who interacts with you while your handler is off taking a leak is also susceptible to corruption by being around you. I'm not sure what kind of hand/body lotion is so potent that it draws demons to people around you, but I'd like to know if my local CostCo carries it in bulk, because some of the people I associate with could use some demonic possession. That's karma for you.
At any rate, Qunari culture is huge on this whole idea of "Vaseline Men are (literally translated) dangerous things that need to be bound, shackled, monitored at all times and killed if there's even a chance they may be corrupted". They're really nice gents. However, there's also an immense amount of respect in Qunari culture for the amount of control they need to have to avoid possession. Saarebas are both hated and feared as much as they are pitied and respected for the weight of their burden.
Given the respect and pity felt for mages and their petroleum plight, it is possible that they may be accepted into the Qun as converts- it takes true amounts of self control to avoid succumbing, especially when not shackled like the Qunari's own are. However, they are still dangerous things that have been exposed for an entire lifetime without a true arvaarad to keep them contained. The best a convert mage could hope for would be a quick, painless death, most likely. An interesting situation,though, would be one of a Templar handler and mage converting together. Would the Templar, for the sake of argument, constitute a sufficient handler? Or is only a true arvaarad acceptable? It's something that would be interesting to see played out, at any rate.
In short: the Qunari believe that anyone with magical Vaseline hands is 120% a threat to everyone if they aren't having their leash held, so at best a mage trying to convert would probably be killed to protect anyone else from their corruption.
(Whew, that got long)