The Israel Defence Forces reported that sirens had sounded in northern Israel, near the border with Lebanon, on Friday afternoon.
Israel did not specify what caused the sirens to go off.
Hezbollah, an Iran-backed paramilitary group in Lebanon, has frequently launched rockets at Israel during the ongoing Israel-Hamas war. Hezbollah is classified as a terrorist organization not only by Israel, but also by the US, Germany and several other countries.
"Hamas chose to violate the cease-fire when they chose not to release the women and children," Lerner told DW. "This is a very clear violation. It was absolutely apparent from day one that it was only a matter of time until Hamas decided not to transfer the hostages."
Lerner said the Islamist militant group could have extended the truce by releasing 20 Israeli women and children it is still holding hostage. Hamas, meanwhile, has accused Israel of ending the truce.
On Friday morning, Israel dropped leaflets on the southern Gaza Strip, telling residents in the city of Khan Younis to move further south to Rafah, located on the border with Egypt.
When asked by DW if this means Israel is expanding its military operation in Gaza, Lerner reiterated: "We will seek out and hunt down Hamas wherever they are in the north or in the south."
"Indeed, we've distributed these leaflets in order to get people out of harm's way," he added. "We do intend on operating. We're not giving a specific timeline, but as we proved in the past, when we say we are intend on operating, we do operate."
Lerner denied that Israel in the past has called on Palestinians to move to certain places in Gaza, and then bombed those areas. He accused Hamas of abusing the "civilian arena" and hiding behind civilian infrastructure such as mosques and schools, putting people at risk.