According to this website, the etymology for country is as follows:

country(n.) mid-13c., "(one's) native land;" c. 1300, "any geographic area," sometimes with implications of political organization, from Old French contree, cuntrede "region, district, country," from Vulgar Latin *(terra) contrata "(land) lying opposite," or "(land) spread before one," in Medieval Latin "country, region," from Latin contra "opposite, against" (see contra-). The native word is land.

Also from c. 1300 as "area surrounding a walled city or town; the open country." By early 16c. the word was applied mostly to rural areas, as opposed to towns and cities. Meaning "inhabitants of a country, the people" is from c. 1300.

Count (like counting up numbers) has a different root:

count(v.) 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."

If you're referring to the title of Count:

count(n.1) title of nobility in some continental nations, corresponding to English earl, c. 1300, from Anglo-French counte "count, earl" (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.

So basically no, it doesn't look like there's any connection between between country and count, they're just similar sounding words.