Disclaimer: I am not a differential geometer, so maybe this question does not make sense:
Let $(M^m,g)$ be a Riemannian manifold and $f:M\to \mathbb{R}$ a Morse function. Since $g$ is pointwise non-degenerate, a obtain a vector field by $X_p:=g_p^{-1}(\mathrm{d}f|_p)$. I have two notions of “critical point” and “index”:
The Morse theoric one: $p\in M$ is critical if $\mathrm{d}f|_p=0$. The index $I(p)$ is the negative sign of $\mathrm{d}^2f|_p$
The one for Poincaré–Hopf: $p\in M$ is critical if $X_p=0$. The index $\mathrm{ind}_p(X)$ is the degree of the following map, locally around $p$ (after choosing charts and obtaining a local trivialisation of $TM$): $$\varphi:\mathbb{S}^{m-1}\to \mathbb{S}^{m-1}, q\mapsto \frac{X(q)}{\|X(q)\|}.$$
Obviously, the two notions of being critical are equivalent. What about the indices? They are not the same as $0\le I(p)\le m$, but not necessarily $0\le \mathrm{ind}_p(X)\le m$. In what way are they related?
The only idea I had was the combination of Poincaré–Hopf and Morse theory (correct?): $$\sum_{\text{critical $p$}} \mathrm{ind}_p(X) = \chi(M) = \sum_{\text{critical $p$}} (-1)^{I(p)}.$$