What, if any, is the connection between turkey, the bird, and Turkey, the country?

Talk Turkey, "Jack"... (The Family Man)

You can catch a glimpse here

turkey (n.) Look up turkey at Dictionary.com
1540s, originally "guinea fowl" (Numida meleagris), a bird imported from Madagascar via Turkey, and called guinea fowl when brought by Portuguese traders from West Africa. The larger North American bird (Meleagris gallopavo) was domesticated by the Aztecs, introduced to Spain by conquistadors (1523) and thence to wider Europe. The word turkey first was applied to it in English 1550s because it was identified with or treated as a species of the guinea fowl, and/or because it got to the rest of Europe from Spain by way of North Africa, then under Ottoman (Turkish) rule. Indian corn was originally turkey corn or turkey wheat in English for the same reason.

The Turkish name for it is hindi, literally "Indian," probably influenced by Middle French dinde (c. 1600, contracted from poulet d'inde, literally "chicken from India," Modern French dindon), based on the then-common misconception that the New World was eastern Asia.
After the two birds were distinguished and the names differentiated, turkey was erroneously retained for the American bird, instead of the African. From the same imperfect knowledge and confusion Melagris, the ancient name of the African fowl, was unfortunately adopted by Linnæus as the generic name of the American bird. [OED]
The New World bird itself reputedly reached England by 1524 at the earliest estimate, though a date in the 1530s seems more likely. The wild turkey, the North American form of the bird, was so called from 1610s. By 1575, turkey was becoming the usual main course at an English Christmas. Meaning "inferior show, failure," is 1927 in show business slang, probably from the bird's reputation for stupidity. Meaning "stupid, ineffectual person" is recorded from 1951. Turkey shoot "something easy" is World War II-era, in reference to marksmanship contests where turkeys were tied behind a log with their heads showing as targets. To talk turkey (1824) supposedly comes from an old tale of a Yankee attempting to swindle an Indian in dividing up a turkey and a buzzard as food.
Turkey Look up Turkey at Dictionary.com
country name, late 14c., from Medieval Latin Turchia, from Turcus (see Turk) + -ia.
turkey-vulture (n.) Look up turkey-vulture at Dictionary.com
1823, from turkey + vulture. From 1670s as turkey-buzzard.

I was under the impression that to talk turkey is to talk frankly, plainly, to the point.

No offense intended.

"Speak plainly, get to the point, as in Don't call me until you're ready to talk turkey. This expression allegedly comes from a tale about an Indian and a white man who hunted together and divided the game. When the white man said, "I'll take the turkey and you the buzzard, or you take the buzzard and I the turkey," the Indian replied, "Talk turkey to me." Whether or not this tale had a true basis, the term was recorded in its present meaning by about 1840." - American Heritage® Dictionary of Idioms