A translation by $x^\nu \to x^\nu - \epsilon^\nu$ corresponds to an infinitesimal transformation of the fields, by
$$\phi \to \phi + \epsilon^\nu \partial_\nu \phi$$
as we are performing an active rather than passive transformation. The Lagrangian transforms as,
$$\mathcal{L}\to \mathcal{L}+\epsilon^\nu \partial_\nu \mathcal{L}$$
by substituting $\phi$ into the Lagrangian. Notice the change is up to a total derivative, and hence Noether's theorem is applicable to the symmetry. The conserved current density is given by,
$$j^\mu = \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}X(\phi)-F^\mu(\phi)$$
where $X=\delta\phi$ and $F^\mu$ is such that $\partial_\mu F^\mu=\delta \mathcal{L}$ infinitesimally. For our case, we obtain the symmetric stress-energy tensor (analogous to that of general relativity),
$$T^\mu_\nu=\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} \partial_\nu \phi - \delta^\mu_\nu \mathcal{L}$$
where the Kronecker delta is raised with the Minkowski metric. The current satisfies, $\partial_\mu T^{\mu}_\nu = 0$, and the corresponding Noether charge,
$$E=\int \mathrm{d}^3 x \, T^{00}$$
is the total energy of the system, whereas,
$$P^i = \int \mathrm{d}^3 x \, T^{0i}$$
is the $i$th component of the total momentum of the field, where $i=(x,y,z)$ only. A caveat: the stress-energy tensor derived by Noether's theorem is not always symmetric, and may require the addition of a term which satisfies the continuity equation, and ensures symmetry in the indices.
Alternate Method
Recall to obtain the Einstein field equations in general relativity, we may vary the Einstein-Hilbert action,
$$S\sim \int \mathrm{d}^4 x \, \sqrt{-g} \, \left( R + \mathcal{L}\right)$$
Similarly, in quantum field theory, we may promote our Minkowski metric to a generic metric tensor, thereby replacing the kinetic term of the Lagrangian with covariant derivatives. Up to some constants, the stress-energy tensor is given by
$$T^{\mu\nu} \sim \frac{1}{\sqrt{-g}} \frac{\partial (\sqrt{-g}\mathcal{L})}{\partial g^{\mu\nu}}$$
evaluated at $g_{\mu\nu}=\eta_{\mu\nu}$, which is precisely the definition we implement when obtaining the Einstein field equations for general relativity.
You are correct in your interpretation that Weisner's method is geometric in nature: it is a method for finding generating functions for special functions using representation theory of Lie groups and Lie algebras. And as you know, Lie groups play an enormous role in modern geometry, on several different levels. Lie groups are smooth differentiable manifolds, that is, roughly, a type of manifold that is locally similar enough to a linear space to allow one to do calculus.
Lie and others showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. The initial application that Lie had in mind was to the theory of differential equations: on the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations. However, the hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled.
Thus, it is correct that you can take
the other special functions mentioned in this paper, obtainable as linear combinations of the conformal symmetries of the Laplacian (expressed as lie algebra elements), and obtain their solution analogously to how Bessel is solved below.
because of how these functions are defined.
However, there is no general method because this is not valid in general for arbitrary ordinary differential equations.
Best Answer
Noether's (first) Theorem is really not about Lie groups but only about Lie algebras, i.e., one just needs $n$ infinitesimal symmetries to deduce $n$ conservation laws.
Lie's third theorem guarantees that a finite-dimensional Lie algebra can be exponentiated into a Lie group, cf. e.g. Wikipedia & n-Lab.
If one is only interested in getting the $n$ conservation laws one by one (and not so much interested in the fact that the $n$ conservation laws often together form a representation of the Lie algebra $L$), then one may focus on a 1-dimensional Abelian Lie subalgebra $u(1)\cong \mathbb{R}$.
In the context of field theory, there should be Lie algebra homomorphisms from the Lie algebra $L$ to the Lie algebra of vector fields on the field configuration space (so-called vertical transformations) and to the Lie algebra the vector fields of spacetime (so-called horizontal transformations).
That the action functional $S[\phi]$ possesses a symmetry (quasisymmetry) means that the appropriate Lie derivatives of $S$ wrt. the above vector fields should vanish (be a boundary term), respectively.
Note that the Noether currents & charges do not always form a representation of the Lie algebra $L$. There could e.g. appear central extensions, cf. this and this Phys.SE posts.