I'm thinking that the front discs are severely worn or there's a hard, crystallized spot on the brake disc which is slightly prominent and making noise when the pads are slightly contacting the disc. Could be the same thing in the rear drum brakes if the noise is coming from the rear. I'd recommend a brake inspection, the brakes may be due for service. If it appears that the discs are worn out or faulty, it's better to replace them (and the pads) than to try to machine a hard spot out, the cost is not much different if you look around for a good price on new discs and hard spots mean that the metal has crystallized and no good any more.
If you want to find a Mazda specialist, here's a directory link for you:
I personally recommend finding a Mazda specialist rather than, let's say, a muffler shop to do the job. Independent specialists use better parts and are more savvy.