count (n.1)

title of nobility, c. 1300, from Anglo-French counte (Old French conte), from Latin comitem (nominative comes) "companion, attendant," the Roman term for a provincial governor, from com "with" (see com-) + stem of ire "to go" (from PIE root *ei- "to go"). The term was used in Anglo-French to render Old English eorl, but the word was never truly naturalized and mainly was used with reference to foreign titles.

In ancient Rome and the Roman empire, [a comes was] a companion of or attendant upon a great person; hence, the title of an adjutant to a proconsul or the like, afterward specifically of the immediate personal counselors of the emperor, and finally of many high officers, the most important of whom were the prototypes of the medieval counts. [Century Dictionary]

count (n.2)

early 14c., "a counting, a calculation," also "an account of money or property," from Anglo-French counte, Old French conte, from conter (see count (v.)).

count (v.)

mid-14c., from Old French conter "add up," but 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." Related: Counted; counting. Modern French differentiates compter "to count" and conter "to tell," but they are cognates.