Count is a weird word. Wait…I think that’s my beginning for all of these. I mean, it’s true, but damn it’s repetitive.
Count—like you do with numbers—showed up in the mid fourteenth century, while the title count showed up in the early fourteenth century. They aren’t related at all. Number count comes from the Old French conter, add up, and the classical Latin computare, calculate. And yes, that’s where compute is from. The com- means with and putare means think, which mostly makes sense, although I'd like to know what happened to the P. Anyway, the title count is from the Anglo French counte and classical Latin comitem, which is…count. It’s what they called some heads of state back then. And it’s also a combo word—again, the com- is from com and it means with, while the -item comes from ire, to go. No, I don’t get that one at all.
And then there’s counter and counter—yes, two counters. One where you do business, and one that means anti-. The first one showed up in the mid fourteenth century from the Old French contouer/comptoir (so they did have an M there at some point). That comes from the Medieval Latin computatoium, a place of accounts, and that then comes from computare. The other counter showed up in the late fourteenth century from the Old French/Anglo French countre, which was then taken from the classical Latin contra, or against.
What about other words with count in them? you might ask. Shut up, I said might. County showed up in the early fourteenth century from the Anglo French counte…okay, it’s getting a bit repetitive. But that’s because county comes from count, because apparently a count rules over a county. And country? Is that related? Ha ha, no. Not even a little. It showed up in the mid thirteenth century from the Old French contree. That comes from the Vulgar Latin terra contrata, land opposite of/land before. Oh, and contrata? From contra. So country comes from opposite. Really.
If you see count in pretty much anything else (counterattack, counterpart), it’s from the opposite counter. But there are other number count words. There’s account, which showed up in the early fourteenth century. The a- comes from ad-, to, and the count can be traced back to computare. There’s also constable, which is related to the title count—it’s actually a mix of that word and stable. You know, like for horses. Finally there’s countenance. Which comes from none of these. It’s related to contain.
TL;DR: Nothing but coincidence.