Recipes such as chilli are normally prety versatile, so you can add/take out what you need.
I think quinoa has a stronger flavour than bulgur wheat (also I haven't had bulgur for ages so I'm not sure I'm remembering the right thing!) so it will alter the taste a little, but if you like quinoa anyway then I'm sure it won't matter too much.
However, I think bulgur is used to give a more "meaty" texture to the chilli, so you may end up with a more sloppy/liquidy chilli when using quinoa. I also think the bulgur may absorb more water than quinoa so thicken the chilli a bit more so you may want to experiment with adding less water than asked for in the recipe.
If I were you I would probably try another recipe then just serve it on top of quinoa if you don't like bulgur, but sometimes it's quite fun to experiment with recipes so you could also give it a go using quinoa - I'm sure it'll taste good even if it doesn't turn out exactly as the recipes states
I believe the bulgur is being used as a ground meat substitute there. Quinoa won't replicate the texture of ground meat the way the bulgur will, but if you just want chili with some quinoa in it, there's no reason why you can't sub.