Probably not. A fallow deer (scientific name Dama dama) is simply a kind of deer which occurs in Europe.
My dictionary tells me that "fallow" can be a red-yellow color, and that that is where the name of the deer comes from.
(Before now, I had only heard the word fallow used to describe fields which have no crop planted on them.)
Positing the "fallow doe" is slang for a pregnant woman takes the romance & mystery out of the song. The expression probably doesn't occur anywhere else in literature, otherwise someone would have written a dissertation on it. ----------- I want to know about these two lines of the song:
She got him up and upon her back and buried him in earthen lake.