The hook length $h_{ij}(\lambda)$ counts the number of boxes directly below or directly to the right of box $(i,j)$. (I picture the Young diagram of $\lambda$ as having the corner $(0,0)$ located in the upper left.) With this in mind, the alternating sum $\sum_{(i,j)\in \lambda}(-1)^{i+j}h_{ij}(\lambda)$ can be expanded as a signed sum over all the boxes in $\lambda$. We see that a box $(i,j)$ contributes $+1$ if both $i,j$ are even, $-1$ if both $i,j$ are odd, and $0$ otherwise. Using this description we see that
$$\sum_{(i,j)\in \lambda}(-1)^{i+j}h_{ij}(\lambda)=\left\lceil \frac{\lambda_1}{2}\right\rceil-\left\lfloor\frac{\lambda_2}{2}\right\rfloor+\left\lceil \frac{\lambda_3}{2}\right\rceil-\left\lfloor\frac{\lambda_4}{2}\right\rfloor+\cdots$$
where $\lambda=(\lambda_1, \lambda_2, \dots)$.

Now we can use a wonderful formula due to Boulet (Theorem 1 in "A Four-parameter Partition Identity") which tells us (after the appropriate specialization)
$$\sum_{\lambda}x^{\sum_{(i,j)\in \lambda}(-1)^{i+j}h_{ij}(\lambda)}q^{|\lambda|}=\prod_{k\geq 1}\frac{(1+xq^{4k-1})(1+xq^{4k-3})}{(1-q^{4k})(1-xq^{4k-2})^2}$$
I will call this two variable generating function $H(x,q)$. The generating function $F(q)=\sum_{n\geq 0}f_nq^n$ is given by
$$F(q)=\left[\frac{\partial}{\partial x} H(x,q)\right]_{x=1}$$
therefore we start by computing
$$\frac{\partial}{\partial x} H(x,q)=H(x,q)\sum_{k\geq 1}\left(\frac{q^{4k-1}}{1+xq^{4k-1}}+\frac{2q^{4k-2}}{1-xq^{4k-2}}+\frac{q^{4k-3}}{1+xq^{4k-3}}\right)$$
and finally setting $x=1$ gives
$$F(q)=\left(\prod_{k\geq 1}\frac{1}{1-q^k}\right)\cdot\left(\sum_{n\geq 1}A_nq^n\right)$$
where $\sum_{n\geq 1}A_{n}q^n=\sum_k\left(\frac{q^{4k-1}}{1+q^{4k-1}}+\frac{2q^{4k-2}}{1-q^{4k-2}}+\frac{q^{4k-3}}{1+q^{4k-3}}\right)$. If we expand this last sum, the terms simplify a little bit to give $\sum_{n\geq 1}A_n q^n=\sum_{k\geq 1}\frac{q^{2k-1}}{1-q^{2k-1}}$.