Do you "call an audible" when you "play it by ear"?

Nope, not related.

"Call an audible" is a term from American Football. It refers to the Quarterback changing the play at the last minute based on how he sees the defense lining up. He will call out the play change vocally to his team-mates while they are already lined up for the play. So it might be used to mean "make a change at the last minute based on the circumstance."

"Play it by ear" is a musical phrase that contrasts against reading from sheet music, so might be used where there is no plan, just acting based on the dynamics of the situation.

But they are used in similar situations when used colloquially. Another synonyms would be "fly by the seat of your pants"

I agree with those on the historical sources of these, but I'm surprised the idea that they are the same. I actually would say the nuance of the two phrases is a bit different and would indicate 2 different sources. When you “call an audible “ the following is implied, 1) there was an existing plan 2) it became obvious this plan was a distinctly worse option than the new plan.

Playing it by ear does not have that connotation.

In other words, the following works only with calling an audible. “We can book the 2 star hotel, but I know you’ll take one look and Call an Audible.”
Meaning you will find it unacceptable.

In contrast, if you were playing it by ear with the 2-star it could be just because you weren’t really planning, ie it was more about having no plan.