I think the paper "A canonical factorization for graph homomorphisms", Barry Fawcett, Can J. Math. 29 (4), 1977, 738-743, answers the question.

Theorem 3 states that in $\textbf{Grph}$, strict epimorphisms are the same as extremal epimorphisms, which are the same as "full epimorphisms", meaning morphisms that are surjective on vertices and edges. The paper doesn't mention regular epimorphisms, but $\textbf{Grph}$ has pullbacks, which I think means that strict epimorphisms are the same as regular epimorphisms.

It's not strictly relevant to this question, but I can't resist mentioning this related paper by the same author about epimorphisms in the category of planar graphs, whose main result will bring a smile to the face of any categorically inclined mathematician.

irreflexiverelations) is not a topos or quasi-topos and is not a particularly nice category. $\endgroup$ – Todd Trimble♦ May 12 '15 at 11:50