No, for God is a spirit, a purely spiritual substance, and does not breathe. The expression is only a human way of putting things. The soul is a spirit, and is called the breath of God merely because caused or created by God in its spiritual or breath-like nature.
Existence and nature of the soul, Question 31 from Radio Replies Volume I Copyright © 1938 (by Rev. Dr. Leslie Rumble, M.S.C. and Rev. Charles Mortimer Carty).