Can I get [0 0] from Categorical Labeling in CNN's?
That should not be possible.  Your "garbage" would be a third class, requiring labels of [1 0 0], [0 1 0], and [0 0 1].
Very simply, the model you've described will return one of two categories, whichever has a higher rating in your final layer.  This happens whether the input values are 0.501 and 0.499, or 0.011 and 0.010 with a large "not sure" portion.  If you don't explicitly code "not sure" into your model, then that portion of the decision will not be considered in the classification.
From what I understand from keras labeling, one hot encoding does not permit the values to be [0 0]? is this assumption correct?
We are trying to classify 2 classes and we want to be able to detect garbage when a garbage image is fed. However, it always detects either 
[0 1] or [1 0]. Is it possible to get [0 0] as a label without introducing a class the will handle the garbage or no?
So basically, can the CNN predict it to be something else if its not the 2 classes?