$M$ complete Riemannian manifold, $\gamma:[a,b]\to M$ is a geodesic, $p=\gamma(a),q=\gamma(b)$. We say q is conjugate to p along $\gamma$ if there is a Jacobi field $J$ along $\gamma$, st $J\neq0,J(a)=J(b)=0$.
But in John Lee's book , he used the definition where $J(p)=J(q)=0$
So is there any counter example of Lee's definition: There is a geodesic segment $\gamma:[a,b]\to M$ that cross the initial point $p$ more than one time, and there is a Jacobi field $J$ along $\gamma$, st $J\neq0, J(a)=J(b)=0$, but also exist $a<a'<b$ st $p=\gamma(a')$ and $J(a')\neq0$.
First, it should be clarified that I didn't write $J(p)=J(q)=0$. That wouldn't make sense, because the domain of a Jacobi field $J$ is a set of real numbers, not a set of points in $M$.
What I did write was
More specifically, what this means is that there are some parameter values $t_0,t_1$ in the domain of $\gamma$ such that $\gamma(t_0)=p$, $\gamma(t_1)=q$, and $J(t_0) = J(t_1)=0$ (while $J$ is not identically zero). (Note that this is essentially identical to the definition that Wikipedia gives, for what it's worth.)
So in the situation you described, $p$ and $q$ would definitely be conjugate along $\gamma$.
That situation can arise. For example, let $M$ be the lens space $L(3;1)$ (the quotient of $\mathbb S^3\subseteq \mathbb C^2$ by the $\mathbb Z/3$ action generated by $z\mapsto e^{2\pi i /3}z$), with the metric inherited from the round metric on $\mathbb S^3$. The geodesics on $M$ are just images of great circles on $\mathbb S^3$; explicitly, with unit-speed parametrization they are $\gamma(t) = [e^{i t}z_0]$, where brackets denote equivalence classes under the $\mathbb Z/3$ action.
There are Jacobi fields along each such geodesic that vanish at $t=0$ and $t=\pi$, so $\gamma(\pi)$ is conjugate to $\gamma(0)$ along each such geodesic. But the geodesic also passes through the same point $\gamma(\pi)$ at time $\pi/3$ (because $[e^{\pi i/3}z]= [e^{\pi i}z]$), and the Jacobi field does not vanish at that parameter value.