Travellers from Slovakia can show border control a printed certificate confirming they have been vaccinated against Covid on the border in Austria. After applying for such certificate at korona.gov.sk, the state will email it to the vaccinated person.
It is also possible for travellers to present the certificate as a screenshot on their mobile phone. Both options do not cost anything.

From late June, the borders of all member states of the EU should be open to vaccinated people.
The Investment Ministry and Slovak state company Slovakia IT, however, developed a third option for proving one has been vaccinated, has overcome the disease or has tested negative for the coronavirus at the border. It should be available from June 26, the Sme daily reported.
Travellers should be able to show the certificate in the new mobile app GreenPass at the border by the end of June. The state has paid more than €60,000 for the app so far, which is three to six times more than price of similar apps.
App that is not necessary
“We do not need an app for that,” said IT expert Ján Suchal from Slovakia.Digital. “The certificate may be photographed on one's phone or accessible via e-mail,” he explained.