Lebanese Prime Minister Najib Mikati said Lebanese officials would "continue to work towards reaching a ceasefire."
The man wounded was an Indonesian soldier, a security source said. A vehicle was damaged, UNIFIL said.
Hezbollah claimed responsibility for a number of strikes along the Israeli-Lebanese border on Wednesday.
By initially subjecting Hezbollah to targeted strikes on “lesser” targets that they can manage, Israel can progressively escalate its operations to target higher-value objectives and so forth.
US envoy Amos Hochstein visited Lebanon for talks with senior officials and will also speak with Israeli officials to see "if this is the right time" to try to resolve the border issue.
Senior White House adviser Amos Hochstein said it was "natural" to look into the issue following the delineation in 2022 of the maritime border between the two countries.
Israel’s ambassador to the UN Gilad Erdan told The Jerusalem Post he was concerned that passage of France’s proposal “might damage severely UNIFIL’s ability to fulfill its mandate.”