We study Gorenstein flat objects in the category Rep (Q, R) of representations of a left rooted quiver Q with values in Mod (R), the category of all left R-modules, where R is an arbitrary associative ring. We show that a representation X in Rep (Q, R) is Gorenstein flat if and only if for each vertex i the canonical homomorphism φ i X:⊕ a: j→ i X (j)→ X (i) is injective, and the left R-modules X (i) and Coker φ i X are Gorenstein flat. As an application, we obtain a Gorenstein flat model structure on Rep (Q, R) in which we give explicit descriptions of the subcategories of trivial, cofibrant and fibrant objects.