A friend always writes in her emails that she and her husband are going to get their “hairs cut at a salon”, not hair cuts or haircuts. She grew up in the 1930s in L.A. so perhaps this was normal for that era but it seems awkward. I would never correct her but I would like to know if this is correct English.
That does not sound right to me. In current LA, I'm sure they don't say that. 1930s: I doubt it, but wasn't there.
Possibly she is making a private joke. I have heard people (mostly women) saying that their husband will get his "hairs" cut, implying that he has so little hair left, that when the barber cuts "it", he actually cuts "them" (one at a time). ...[impertinent question: is her hair thinning also?]
"Hairs cut" is not idiomatic. It should be "haircuts".
However, there is a silly joke (one we might call a "Dad joke") that relies on the ambiguous relationship between the uncountable noun "hair" (the part of the body that refers to the quantity of hair on one person's head) and the countable noun "hair" (as in a strand of hair):
Son: Hey Dad, can I have some money to get a haircut?
Dad: How much do you need?
Son: It's $20.
Dad: Well for that price you'd better tell them to cut more than one hair.
Your friend might also be making a joke, or it might be a common expression used by her friends and family.