NFA automata with $\epsilon$ moves
It is true that if $q_0 \in F$ then $\epsilon \in L(A)$. The converse holds for DFAs and for NFAs without $\epsilon$ moves, but doesn't hold for NFAs which are allowed to have $\epsilon$ moves. That is, there is an automaton $A$ whose initial state is not accepting, yet $\epsilon \in L(A)$. I'll let you find an example.