Several Israeli media outlets reported that the moves were made in response to an increased threat from the Iran-backed Lebanese Shiite group Hezbollah, which has a significant presence in Syria.
Hezbollah seeks Lebanon's strength and independence, Sheikh Qassem said, adding that the party's popularity, not military power, enabled it to be represented effectively in the parliament and the government.
Israel had been on high alert that they could be attacked after an Israeli airstrike in Syria killed a Hezbollah militant last week. "Non-essential vehicle traffic should be avoided", an army spokesperson said.
Israel's public broadcast channel Kan said there had been an exchange of fire.
"A little while ago, there were reports of a security incident in the area of Jabal Ros", Avichai Adraee, a spokesman for the Israeli army, said in a statement.
Israel said it repelled an attempt by "terrorists" to penetrate its territory, opening fire on the gunmen just after they crossed the frontier with Lebanon.
The militia rejected the accusations. "Any kind of activity in open areas is forbidden", an army statement said.
IDF Spokesperson Hidai Zilberman said at least some members of the Hezbollah cell fled back to Lebanon, but added: "We don't know their conditions for sure at this point".
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"He urges maximum restraint", the UNIFIL statement said.
Former environment minister Wiam Wahhab tweeted: "I doubt that what happened in South Lebanon today was an operation by the resistance...but it seems that the Israelis are petrified from an operation".
The incident started around 3:00 p.m. local time and the Army ordered all residents of the Upper Galilee region to remain indoors until further notice. It was their first cross-border clash for years.
"We are constantly monitoring what is happening on our northern border. We stepped out to the balcony and saw thick smoke rising up near Haramoun Mountain", he said.
Israel sees the existence of Hezbollah and its ally Iran in Syria as a tactical risk and has actually installed numerous raids on Iranian- connected targets there.
"Lebanon and Hezbollah will bear the responsibility for any attack from Lebanese territory", Israeli Prime Minister Benjamin Netanyahu said in broadcast remarks, shortly before the reports of the incident.
The deputy chief of Lebanon's powerful Hezbollah movement on Sunday dismissed the prospect of an escalation of violence between the Iran-backed movement and Israel despite increased tensions in the last week. First, we will not allow Iran to entrench militarily on our border with Syria.
Hezbollah denied all Israeli reports regarding the clashes or gunfire exchange with the Zionist Regime on Monday. Third, the IDF is prepared for any scenario.