Thanks for your answer but i unfortunately i am not familiar with these terms. However, i think i know how to work it out now:

Wlog we may assume that

. Let

and

the associated graphs of G and H. Define the graph

by

and there is an edge between (a,x) and (b,y) iff there are edges from a to b in

and x to y in

having the same label. Since G and H are f.g. the graphs

and

are finite graphs, thus

is also a finite graph whose fundamental group is

. Hence the intersection is f.g., qed.

What do you think of that proof?

Greetings

Banach