How can I tell if a system has a quadrupole moment? We know that gravitational waves are emitted (at least in GR) when the system has a time-varying quadrupole (or higher) moment. My question is
Is it possible to easily tell (e.g. just by looking) if a system has such moments?
Is there some kind of symmetry to look for that lets you know that the system will emit such radiation?
 A: I don't think it possible to easily tell, but here is how you might approach this.  The multipole moments $T_{\ell, m}(t)$ of the energy density $T^{00}(t, \mathbf x)$ of a source can be written on surfaces of constant time in terms of spherical harmonics as follows:
$$
  T_{\ell m}(t) = \int d^3x \, Y^*_{\ell, m}(\theta, \phi)r^\ell\,T^{00}(t, \mathbf x)
$$
Notice that if we write the energy density in terms of spherical harmonics;
$$
  T^{00}(t,\mathbf x) = \sum_{\ell, m}c_{\ell, m}(t,r)Y_{\ell, m}(\theta, \phi)
$$
then performing the integration in spherical coordinates and using orthogonality of spherical harmonics (I can give details if you want), we find
$$
  T_{\ell, m}(t) = \int dr\, r^{\ell+2} c_{\ell, m}(t,r)
$$
Therefore, we see that if certain of the coefficients $c_{\ell, m}$ in the expansion of the density in terms of spherical harmonics vanish, then the corresponding multipole moment vanishes.  The monopole moment corresponds to $\ell = 0$, the dipole moment to $\ell = 1$, and the quadrupole moment to $\ell = 2$.
So say, for example, that the energy density has spherical symmetry, namely
$$
  T^{00}(t,\mathbf x) = f(t,r), \qquad r = |\mathbf x|
$$
Then its expansion contains no spherical harmonics with $\ell>0$, so all multipole moments other than the monopole moment vanish.  In particular, there is no quadrupole.
So the best advice I can give is to look at the pictures of spherical harmonics, and attempt to get some intuition for when certain energy distributions will contain them in their expansions.
I hope that helps!  Let me know of any mistakes and/or typos.
Cheers!
A: The simplest way to construct a quadrupole is to have two identical dipole displaced by some distance opposed to each other,  there is a quadrupole.
In some sense, a dark mode dipole (with dipole canceling each other in far field, but not canceling in  near field), gives you a quadrupole. An octupole is constructed by placing quadrupoles in a similar fashion.
Or actually, as long as the source is smeared in a finite volume and oscillating with time, you will have all multipole modes up to arbitrary order (just increasingly vanishing). This is the result of multipole expansion. 
