How can I express the time between late 80s and beginning of 90s (somewhere around 1988-1992)?

"Establishment of Political Parties in Georgia (1988–1992)"

Oliver's answer seems best and most precise, but if you are looking for something more vague, "conjunction" is awkward usage.

If anything, you could say "at the turn of the 1990s"; we don't talk about the turn of decades as often as we do the "turn of the century" but it's as valid and easily understandable.

See also: https://ell.stackexchange.com/questions/78220/about-the-period-of-from-the-turn-of-the-decade

"Establishment of Political Parties in Georgia (in the late '80s and early '90s)"

Depending on the political sensitivities of your intended readership you could consider some historical landmark such as ‘end of cold war’ Or ‘collapse of communism in Europe’?