As John Rennie says the hydrogen atom is a case where the electron is in the s shell, which means it has no angular momentum. We can think of the electron if it were measured as being at a point above the proton, where the electron as a wave would then spread into a spherical shape around the proton. One might imagine a sort of Zeno machine that keeps the electron wave function reduced so it remains at a point, or a set of points that hop around. If the Zeno effect is a set of measurements occurring at time intervals much shorter than the spreading time of the electron wave function then this hopping of the point can be minimized.
Atomic physics is a bit complicated. The wave function has a radial and angular part to it. The angular part defines the shells, s, p, d, f etc. These are also a series in multipole moments, s = spherical, p = dipolar, d = quadrupolar and so forth. So we need to consider the quadrupolar terms. This occurs with atoms in the transition metals, with scandium being the first. This would have a single electron in the outer d shell.
How would one then consider gravitational radiation produced by the quadrupole motion of a particle? This is a sketch of how to look at weak field gravitational radiation. A full treatment is a bit longer. We start with the metric $g_{\mu\nu}~=~\eta_{\mu\nu}~+~h_{\mu\nu}$ where $\eta_{\mu\nu}$ is the flat Minkowski background metric and $h_{\mu\nu}$ is the perturbation on that. We further look at the traceless components of this metric perturbation which we label as $\bar h_{\mu\nu}$. These traceless metric components have two non-zero terms $h_{ii}~=~A_+(t,r)$ and $h_{ij}~=~A_\times(t,r),~i~\ne~j$ for the components indicies $i,~j$ running over two spatial dimensions. These traceless metric components obey the inhomegenous wave equation
$$
\square\bar h_{\mu\nu}~=~-\frac{16\pi G}{c^4}T_{\mu\nu}
$$
Now set $G/c^4~=~1$ for simplicity. These metric coefficients are then written according to a Green's function with
$$
\bar h_{\mu\nu}~=~-16\pi\int G_{\mu\nu}^{\alpha\beta}(t,{\bf r},t{\bf r}')T_{\alpha\beta}(t,{\bf r}')
$$
We now expand the Green's function according to spherical harmonics and consider the quadrupolar terms the traceless metric terms are then approximately
$$
\bar h_{\mu\nu}^{\alpha\beta}~=~-4\int d^3r \frac{Q_{\mu\nu}(t-|{\bf r}-{\bf r}'|,{\bf r}')}{|{\bf r}-{\bf r}'|}.
$$
That is the classical theory. We want to quantize this. We then have a wave function(al) of the form $\Psi[h]$. To make this simple we then consider this wave function(al) as expanded according to a radial and angular part. The $\frac{1}{|{\bf r}-{\bf r}'|}$ part of the metric means we will have a radial part similar to the Laguerre polynomials in atomic physics, and we then consider the quadrupole term as giving the Legendre polynomial term $Y_\ell^m(\theta,\phi)$ for $\ell~=~2$. We may then proceed with an atomic physics calculation, which below I will only gives a few pointers on.
The stress-energy term $T^{00}~=~\rho$ the energy density, is then expressed as $T^{00}~=~\hbar\omega/volume$. The coupling term for gravitation is $G/c^4$ and so there is the factor $G\hbar/c^4$ associated with the perturbation of the d shell due to gravitation. An atomic transition due to the emission of a graviton would be the emission of a spin-$2$ particle and the transition in $\ell~=~2$ to $\ell~=~0$, so the entire quadupole term is carried off by the graviton. This would have a coupling term $\sim~G\hbar/c^4$, which is $8.7\times 10^{-79}m-s$. This is very small.
Since this is computed for a quadrupole moment or the d shell, one would either have to work with excited hydrogen atoms that remain in that state long enough to perform measurements, or one has to work with a transition metal such as scandium. In the first case this would be tough to measure the perturbation of the d shell by gravitation in a time within the transition time for the atom to relax to the s-shell. If one works with a transition metal that problem is replaced by the fact the underlying electronic configuration will have a lot of complexity that needs to be computed the very high order in perturbation theory, such as Hartree-Fock method. Either way this is a tough call, but not absolutely impossible. I think working with higher Rydberg atomic states of a hydrogen atom would be most likely bear fruit.
The physics that is most likely relevant will then be the perturbation of the d shell by gravitational physics. This will be very small. There could of course be a transition that produces a soft graviton as presented by Weinberg, but the coupling is very small and the probability of such a transition in any reasonable period of time extremely small.
