Unless that rock came from somebody's house who also has dragons with mites...it won't be the source of the mites.
Nearly all species of mites are very species specific. Thus snake mites are not likely to thrive on lizards and vise versa (they may bite and attempt to feed on each other but wont' survive long as the blood won't be compatible.)
Provent a Mite or Reptile Relief are save and effective Mite killers you can use. Both can be sprayed over the entire enclosure, just remove the animal and the water bowl first.. let dry then put the animal back in. With PaM treat again in a month, repeat another 1-2 times for heavy infestations.
Cages and furniture should all be washed, and sterilized (hot soapy water to wash, then bleach solution (1/10 bleach/water) soak for 20 minutes, rinse, rinse, rinse..air dry in ventilated area. For wood cage furniture, same treatment...may even be necessary to just toss out and replace.
Soaking your bearded dragon in water with Betadine (mix till water looks like weak tea) for about 10 minutes (good ideal to soak in regular water, let the dragon drink and maybe defecate...then take it out, put fresh water, add the betadine and put the dragon in for the 10 min. soak). This will disinfect the bites and help prevent infections of the bites themselves.
Also a good ideal to keep the cage simple, with basking spot, paper towel and hide.. for the next 2-3 months...makes it easier to clean, less places for mites to lay eggs.
Use treatment for all bearded dragons you have, and other reptiles in the same room and near anywhere your dragon roams, if you take them out. Mites only 'bite' and drink blood in one stage of their life cycle..roaming mites can go far before they mate and lay eggs...and the baby mites can roam far before they need to feed.
For both Provent a Mite and Reptile Relief, follow the directions on the can.
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PHLdyPayne