The overlap is that the verb "bokeru" and its root "boke" can be used to describe someone losing their mental faculties e.g. through age or disease such as Alzheimer's, and by extension it can be used as an insult to mean "stupid" as well. But etymologically there is no connection.
Thank you for the Wiktionary here. We can see that the Kanji for baka (馬鹿) are horse and reindeer. Non-sequitur combinations in Japanese usually tend to hint at „ateji“, or a phonetic approximation using Kanji. Chinese has a word for the same Kanji as well, but it does not mean idiot as far as I can tell.
Boke can mean either 惚, which is often used to describe someone who has gone senile (ぼけている). Sometimes also for stupor-like situations such as a heavy jet lag. Or, as 暈け, describing something not-sharp and out of focus.
Funnily enough, in English we use sharpness for intellect as well, so ESL learners might think that natives think of their brains as swords? There are enough blade-like metaphors for our intellect, come to think of it.
In more common Japanese, when something has gone out of focus such as the TV, ぼやける is used. One could think that this is related to ぼけ? But one can never be so sure. Unique two-syllable combinations in Japanese are not so numerous and factoring in sound shifts over time, different potential etymologies, one really needs to investigate this for quite a while and maybe never get to the truth. This warrants more investigation!
Supposedly, if I'm remembering last discussion with a japaneese speaker correctly, the same stem is used for "blur", or "blurry" (bokeh, bokeshi).
Which is kind of interesting parallel here