No one will be able to answer your question without a time frame. Additionally, a name would help immensely in tracking the movements of your ggm.
Who informed you she illegally crossed the border and what is the source of their information? What is the nature of the deportation records from Quebec? Is it an addendum to an immigration record or some other type of record?
It has been general practice that an immigrant refused entry is taken into custody at the port of entry and remains in custody at that location until the steamship company makes arrangements for her return to the port of embarkation.
There is a difference between being refused entry and having been deported. It is important to know the sequence of events that got her from the foreign port to the Canadian port, where she was examined by immigration, the results of that examination, and the manner and place of her crossing the border into the US.
Whatever additional information you can provide will help in unraveling the mystery.