It comes after a significant rise in cases of coronavirus being imported from the Mediterranean country, the Scottish Government said.
The measures will take effect at 4am on Thursday.
It came as the Welsh Government announced that passengers on a flight from Zante to Cardiff International Airport tonight will be asked to self-isolate for 14 days due to the risk of spreading coronavirus.
Scottish Justice Secretary Humza Yousaf said: “We are in the midst of a global pandemic and the situation in many countries can change suddenly. Therefore, people should think very hard before committing to non-essential travel abroad.
“With Scotland’s relatively low infection rate, importation of new cases from Greece is a significant risk to public health.
“We continue to closely monitor the situation in all parts of the world and base the decisions we make on the scientific evidence available.
“Regular discussions continue with the other three governments in the UK.
“Requiring travellers arriving from a non-exempt country to quarantine for 14 days on arrival is vital to helping prevent transmission of the virus and to suppress it.”
The Scottish Government said evidence of virus importation, especially from the Greek islands, has led to the country being removed from the exemption list on public health grounds.
It is believed prevalence of Covid-19 in Greece remains lower than 20 per 100,000, but a number of cases in Scotland can be traced back to travel to the Mediterranean country.
Countries tend to be added to the quarantine list when the figure rises above 20 per 100,000.
Scotland’s chief medical officer Gregor Smith said: “There is a compelling public health risk around importation of the virus, especially given the number of imported cases linked to the Greek islands.
“The flow of travel between Scotland and Greece, and the behaviour we have seen from some of those travellers, means that on public health grounds there is a strong case – supported by public health directors – to remove Greece from the exemption list.”
All international travellers coming into Scotland, apart from a very limited number of individual exemptions, must complete a passenger locator form.
Those who do not complete it and present it when asked on arrival may be fined £60.
This can be doubled for each subsequent offence up to a maximum of £480.
Failure to comply with the requirement to quarantine may result in a fine of £480.
In a written statement, Welsh health and social services minister Vaughan Gething said: “In the last week our NHS Wales Test Trace Protect service has identified multiple separate clusters linked to the Greek island of Zante which currently has no returning quarantine restrictions.
“There are currently six clusters amounting to over 30 cases in the last week from four flights, of which two of these flights landed in England.
“There are concerns from our public health teams that the current advice and control measures for returning travellers are insufficient.
“Unfortunately our consultants in communicable disease control have several examples of Covid-19 positive travellers who have not self-isolated on return to Wales. That is a real concern for all of us.
“Public Health Wales have expressed direct concerns about the public health risk posed by passengers returning to Cardiff this evening from Zante.
“I share that concern. It is almost certain that travellers returning to Wales from areas of higher Covid-19 incidence will lead to further seeding of infections within Wales.
“Travel into Wales from mainland Europe drove the first wave of Covid-19.”
With reporting by Press Association