Lefschetz Fixed Point Theorem:
For a compact triangulable space $X$, and a continuous map $f:X\rightarrow X$, we have $$\sum_i(-1)^i\mathrm{Tr}(f_*|H_k(X,\mathbb{Q})=\sum_{x\in\mathrm{Fix}(f)}\mathrm{index} _x f$$
Poincare-Hopf Index Theorem:
For a compact orientable differentiable manifold $M$, and a vector field $v$ on $M$ with isolated singularities, we have $$\sum_i(-1)^i\dim H_k(X,\mathbb{Q})=\sum_{x\in\mathrm{Sing}(v)}\mathrm{index}_xv$$
Observing the formal similarity between these formulae and the compatibility of the conditions, I came up the idea that they are connected.
For one possible connection, I imagined that we can construct a map $f_v:M\rightarrow M$, by letting the points flow along $v$ (in a sufficiently short time?). Then this map is very similar (homotopic?) to $\mathrm{id}_M$, so trace is dimension. And by construction the two index should agree. So we can view the latter as a corollary.
But I was not able to realize this. Can anyone tell me how to do it? Or it can be found in some texts?
You can look in the text of Guillemin and Pollack, Differential Topology. You'll find precisely the answer to your question (for manifolds).