Yes. Let $H$ be a finitely generated subgroup of $F$. Let $[0,a]$ be the largest interval where all elements of $H$ are equal to the identity. Then consider the map that sends $h\in H$ to the $\log_2$ of the (right) slope of $h$ at $a$. This map is a non-trivial homomorphism of $H$ into $\mathbb{Z}$. 

A more fancy (but essentialy the same) proof is this: by Thurston's stability theorem (W. Thurston. A generalization of the Reeb stability theorem. Topology 13 (1974), 347-
352), every group of $C^1$ diffeomorphisms of the closed interval is locally indicable, and by Ghys and Sergiescu, $F$ is a subgroup of the group of $C^\infty$ diffeomorphisms of the interval $[0,1]$ (see E. Ghys and V. Sergiescu: Sur un groupe remarquable
de diffeomorphisms du cercle, Comm. Math. Helv. 62
(1987) 185–239.).