Могут. Вы нарушили закон РК. По последствиям - ну вот например что UK Foreign Office говорит
https://www.gov.uk/foreign-travel-advice/cyprus/safety-and-security#:~:text=The%20Government%20of%20the%20Republic%20of%20Cyprus%20may%20fine%20you,the%20boundary%20with%20the%20north.Even when the crossing points are operating under normal conditions, British and other foreign nationals who have entered Cyprus through the north (such as via Ercan airport) are considered by the Government of the Republic of Cyprus to have entered Cyprus through an illegal port of entry. The Republic of Cyprus does not recognise the self-declared ‘Turkish Republic of Northern Cyprus’ in the north of Cyprus, nor residence permits issued by the administration there. The Government of the Republic of Cyprus may fine you for illegal entry if you cross into the south, or refuse you entry into or exit from the Republic, or prevent crossing at the boundary with the north.