Could a strong acid become a strong base?
Take nitric acid for instance. Reduce the nitrate until ammonia forms. Then using a stronger base than amide ion, you can generate amide base.

Therefore, it is required to make a number of reactions to get a strong base from a strong acid. If you meant that could a strong acid be changed to a strong base by one step reaction, it might be possible with complex big organic or organoboron molecules by replacement of acid functional group with a basic one, but I am unsure about this part.