Determine if there is a path with alternating edge colours in directed graph
Let's construct a new graph $G'$. 

Every vertex $u$ from $G$ will become a pair of vertexes in $G'$, namely $u_{blue}$ and $u_{green}$ (index stands for the color of incoming edge). 

Edge $u_{c_1}\rightarrow w_{c_2}$ in $G'$ exists iff


There is an edge $u \rightarrow w$ in $G$ colored $c_2$
$c_1 \ne c_2$ 


Now it's easy to see that every path in $G'$ corresponds to a color-alternating path in $G$ (and the converse is also true), i.e. color-alternating path from $v$ to $u$ in $G$ exists iff there is path from $v_{c_1}$ to $u_{c_2}$ in $G'$ for some $c_1, c_2$.