No, and I would argue that "the girl is bilingual now" is proper English and the more correct one of the two, whereas "the girl is now bilingual" is a bit awkward.

In English "now" is an adverb of time. Usually adverbs of time are put at the end of the sentence, but they can be placed at the beginning or mid-sentence for different emphasis.

"The girl is bilingual now" is more common than "The girl is now bilingual". Both sentences are correct English but "The girl is now bilingual" emphasizes the change from her being unilingual to being bilingual.

I couldn't find good examples using "now" in different positions, so here are sentences using the adverb of time "later":

• Later Goldilocks ate some porridge. (the time is important)
• Goldilocks later ate some porridge. (this is more formal, like a policeman's report)
• Goldilocks ate some porridge later. (this is neutral, no particular emphasis)

Source of sentences using "later": http://www.edufind.com/english-grammar/adverbs-time/