Can you recover the quantum numbers from just the shape of the spherical harmonic?

Yes, is it possible to read off $ell$ and $m$ from the shape of the spherical harmonic $Y_{ell m}$. The rules are not complicated:

• Number of plane nodes (containing the $z$-axis): $|m|$
• Number of cone-like nodes (around the $z$-axis): $ell -|m|$

There is a nice interactive web page for Visualization of Spherical Harmonics. You can choose values $ell$ and $m$ to get an image showing positive $Y_{ell m}$ as red bumps, negative $Y_{ell m}$ as blue bumps, and also the nodes as thin lines.

Here is the image (with $ell=6, m=3$) corresponding to the example image from your question:

_{(image generated using Visualization of Spherical Harmonics, published by ICGEM Potsdam)}