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For a quantum harmonic oscillator, the ground state in most sources is referred to as $n=0,$ and this state has zero nodes.
For a particle in a box, the ground state in most sources is called $n = 1.$ It has zero nodes.
So why is the ground state called $n=1$ for the particle in a box and $n=0$ for the harmonic oscillator?
Discrete eigenvalues can always be labeled by the integers, so we can always pick up $E_n$ with $n$ starting from $0$ or $1$, it is just a matter of convenction. I suppose this is where your question originates. For the harmonic oscillator the energies are given by $\hbar \omega (m +1/2)$ with $m=0,1,\ldots$ so it is natural to pick $n=m$ starting from $0$.
For the particle in the box, note that the problem is equivalent to a "free" particle with certain boundary conditions (wavefunction zero at the border). In this sense it is no surprise that "momentum" is conserved (people speak of quasi-momentum in this case). In this picture the Hamiltonian is simply
$$H = \frac{p^2}{2m}.$$
If you set the box as in $[0,L]$ you can write the eigenfunctions as
$$\psi_k(x) \propto \sin(kx)$$
where $k$ is this quasi-momentum to be determined. In this way one of the boundary conditions is automatically satisfied. To satisfy the other one we must impose $\sin(Lk)=0$ which in principle leads to $k= m\pi/L$, with $m\in \mathbb{Z}$. However, $k$ and $-k$ single out the same quantum state (a quantum state is an equivalence class of vectors which differ by a phase). Moreover $k=0$ is not allowed because the zero wave-function is not normalizable. Hence we are led to
$$k= \frac{n\pi}{L}, \ \ n = 1,2,\ldots$$
In a sense, physically, this tells us that the state with zero momentum (a state that does not move left or right) is not allowed. This is clearly a quantum mechanical effect (a fixed particle is perfectly allowed classically). You can think of its origin in the Heisenberg's uncertainty principle. Since $H\propto p^2$ we can obviously label the energies with the same label that we use for the momentum (another good notation would be simply to label them with the quasi-momentum itself as in $E_k$ and remember what is the spectrum of $p$, $\sigma(p)$.
I suppose you knew all of this but I don't think you can say much more.
Edit
Let me briefly comment on the number of nodes of the wavefunctions.
For a Hamiltonian of the form
$$ H= \frac{p^2}{2m} + V(x) = - \frac{\hbar^2}{2m} \frac{\partial^2}{\partial^2 x} +V(x) $$
a theorem by Courant and Hilbert (generally called nodal theorem) states that the ground state wave function has no nodes (can be chosen to be everywhere non-negative) and the $n$-th level has precisely $n-1$ nodes. Moreover if the potential goes to infinity as $|x| \to \infty$, then the eigenvalues form a discrete unbounded sequence. I imagine variations of this theorem exist for $\mathbb{R}^d$. So the behavior you observed for the number of nodes is in fact quite general.
As with all questions of the form "why does this convention...", the answer is ultimately "because conventions are arbitrary".
Still, you are correct in noticing that the harmonic oscillator does buck an overall trend (which has instances in the hydrogen atom and the particle-in-a-box problem, among a few others) where eigenstates are numbered as $1,2,3,\ldots$, with integers starting at unity. So what is it about the harmonic oscillator that makes it more convenient to change that trend? (After all, if it wasn't more convenient, why change the trend?)
The reason is simple ─ the $n$th wavefunction of the harmonic oscillator has the form $$ \psi_n(x) = H_n(x) e^{-x^2/2} $$ (modulo constants), where $H_n(x)$ is a polynomial of degree $n$, starting with degree zero (a constant) for the ground state and then moving up with the eigenstate number. The Morse potential, which has a similar numbering scheme, also has this type of polynomial dependence.
This isn't a hard-and-fast rule, though, and it's likely that there are other systems with other types of exceptions. Conventions are tricky things, and they are social objects - it doesn't make too much sense to obsess over why the details settled to where they are.
