We recently adopted a 2 year old rottweiler. On occasion, she uses a shaggy bathroom rug to poop, despite being given opportunity to do her business outside. Does she think this rug is her grass? Is removing the rug the only way to solve this problem?
As your dog has only recently become part of your family, it is possible that she is still feeling unsettled and is getting used to the household routine. As her history is unclear, it could be that she is not accostomed to doiing her business outside. You will just have to keep encouraging her to go to the toilet outside and rewarding her when she does. Being patient is the key here, and make sure you don't ever tell her off unless you catch her in the act of toileting on the mat. It will do nothing to rub her nose in the mat, because she will not associate making a mess with the scolding, and this will only serve to make her fearful.
If the only thing inside that she is doing her business on is the rug, then it probably would be a good idea to take it away, just to encourage her to go outside.
I hope this helps and all the best with the training!