Given a four dimensional Lorentzian manifold $\mathcal{M}$ (a manifold with a metric $g_{\mu\nu}$ in the tangent bundle with signature (-1, 1, 1, 1)), we define a global spatial foliation by a time-like vector field ($n^\mu{}n^\nu{}g_{\mu\nu} = -1$). Is it true that given another global foliation defined by a time-like vector field $v^\mu$ there is (at least one) a diffeomorphism $\Upsilon:\mathcal{M}\rightarrow\mathcal{M}$ generated by a vector field $A^\mu$ such that $v^\mu = \exp(\mathcal{L}_A)n^\mu$, where $\mathcal{L}$ represents the Lie derivative and $\exp(\mathcal{L}_A)$ the exponential map generated by it?
If it is true, is there a good reference for such result?
Here are details for my comment.
The answer to your question (without further assumptions) is negative. Here is an example.
Consider $R^{3,1}$ with the (standard) flat Lorenztian metric $x_1^2+x_2^2+x_3^2-x_4^2$. Let $\Gamma$ be the discrete isometry group of $R^{3,1}$ consisting of translations by vectors with integer coordinates. The quotient $M=R^{3,1}/\Gamma$ is the 4-dimensional torus with flat Lorenztian metric. Next, let $\tilde \nu$ be the (parallel) vector field on $R^{3,1}$ consisting of vectors parallel to the vector $(0,0,0,1)$ and let $\tilde n$ be the parallel vector field defined similarly, using the vector $(t,t,t,1)$, where $0<t<1/\sqrt{3}$ is a small irrational number (it is then also time-like). Then take $\nu$ and $n$ to be projections of the vector fields $\tilde\nu, \tilde n$ to the torus $M$. The normal foliation to $\tilde \nu$ is by horizontal hyperplanes. Each leaf projects to a 3-dimensional torus in $M$, which gives us a foliation of $M$ normal to $\nu$. On the other hand, each hyperplane normal to $n$ projects to a hypersurface which is dense in $M$ (this is an irrational foliation of $M$ normal to $n$). Now, it is clear that there is no homeomorphism $M\to M$ which carries one foliation to the other one.
My guess is that your question has positive answer (but the diffeomorphism might be more complex than the one you are attempting to construct) in the following setting: Both $\nu$ and $n$ are normal to product foliations, defined below.
Definition. A foliation $F$ on $M$ is a product foliation if there exists a diffeomorphism $h: M\to X\times {\mathbb R}$, where $X$ is a compact 3-dimensional manifold, such that $h$ sends leaves of $F$ to submanifolds of the form $X\times \{t\}, t\in {\mathbb R}$.