The frequency of both words being used in the same sentence the past couple days has me wondering about the relatedness. So, I looked up the etymology of both.

• count (verb) late 14c., “to enumerate, assign numerals to successively and in order; repeat the numerals in order,” also “to reckon among, include,” from Old French conter “to count, add up,” also “tell a story,” from Latin computare “to count, sum up, reckon together,” from com “with, together” (see com-) + putare “to reckon,” originally “to prune,” from PIE root *pau- (2) “to cut, strike, stamp.”
• county (noun) mid-14c., “a shire, a definite division of a country or state for political and administrative purposes,” from Anglo-French counte, from Late Latin comitatus “jurisdiction of a count,” from Latin comes (see count (n.1)). It replaced Old English scir “shire.”

So, not at all. Both come from different French and Latin terms. French conter vs counte and Lating computare vs comitatus.

English is weird, yo.