I have the 1-form $$\alpha = pdq-\frac{(p^2+q^2)}{2}dt$$ and the vector field $$v=p\frac{\partial}{\partial q} - q\frac{\partial}{\partial p}.$$ How would I find conserved quantities using Noether's Theorem?
Additionally, is there a difference between finding conserved charges and conserved quantities?
I'm a math student who hasn't taken many physics classes so I am a bit lost with this.
Well, the system is secretly on Hamiltonian form, meaning that there is lurking a symplectic structure $$\omega ~=~ dp\wedge dq$$ in the background, and there is a Hamiltonian $$H~=~\frac{p^2}{2}+\frac{q^2}{2}.$$ Here $$\alpha~=~\mathbb{L}_H~=~pdq-Hdt$$ is the Hamiltonian Lagrangian one-form. If $\gamma: [t_i,t_f]\to \mathbb{R}^2$ denotes a path in phase space, then the corresponding action functional reads $$S[\gamma]~=~\int_{t_i}^{t_f} \! \gamma^{\ast}\mathbb{L}_H.$$ The symmetry-generating vector field $$v~=~-X_H~=~-\{H,\cdot\}_{PB}$$ is the Hamiltonian vector field generated by $-H$.
This means that we can use a Hamiltonian version of Noether's theorem, cf. this Phys.SE post. We leave the details to the reader, but the main answer is that the Hamiltonian $H$ itself is the sought-for conserved charge/quantity.
