I have smaller feet than you, so it can be done!!!!
Burton does make a size 4 in a few of their boot models. Many small sizes are available in "Asian fit only" - not sure what that means. I know nothing about the Q, though.
My boots are Salomon Ivy - size 4 (Mondopoint 22). I love them - high end, very comfortable, with a good level of support. Check out the Salomon line, for sure.
I *believe* Nitro and possibly Forum also make boots in sizes 4 or 4.5. Those sizes are not available in all models, though.
DO NOT go with children's boots. They are low quality and too soft to support your weight.
Suggestion: go to a store that sells ski and snowboard boots and have your feet measured in centimeters (Mondopoint) - that will help give a clearer idea of what size you need, since you'll never find a boot to try on in a store. My true MP size is a 21.5. I wear a MP 22 (Salomon size 4), which is slightly big, but I've luckily had no sore feet as a result.
As far as I know, there are a few tricks you can use to make big boots fit better, but none are permanent fixes and none are as desirable as wearing the correct size. Locate the few boots that are made for us little people, choose the one that fits your riding style and needs the best, take a deep breath, and order them online. I bought mine on Backcountry.com. $200 boots left over from last year, on sale for $85.
Good luck and feel free to PM me if you need to. I've been there - it's a frustrating process!!