M-x describe-input-method RET tex RET will tell you the (somewhat depressing) news: lots of letters can be "hatted" but they are Latin letters only. Input methods are limited in the kind of transformation they do. There is no "hatted beta" Unicode symbol either, so the only way to do it is by writing your notes in LaTeX (or Org mode using LaTex) and produce PDF: then you can write
If you write in Org mode (with LaTeX), you can turn on automatic preview of LaTeX fragments and see the hatted beta as an image overlay in the emacs buffer. Or you can use AucTex and its preview feature.
More details about the Org mode/LaTeX preview method can be found in this answer. More details about AucTex and its LaTeX preview feature can be found in the AucTeX/Preview LaTeX manuals.