This is not accurate, as far as historians currently understand it.

The Israeli Prime Minister, Golda Meir, did reportedly threaten to use nuclear weapons. However, this was not premised on the question of whether or not forces entered "Israel proper", nor was the threat specifically related to Damascus.

Rather, during the war, the Arab states were being supplied and resupplied by the Soviet Union. Israel, recognizing that it had been caught off-guard, also recognized that it did not have the manufacturing capacity to keep up the continued fighting and push back against the invading Arab forces. This greatly worried Israeli leaders, since the "greatest airlift in [Soviet] history" was declared by an Egyptian general to be starting on October 10. In response, the US considered resupplying Israel.

Israel, however, had a way to entice the US, or so the thinking goes on the next part of the story. Secretary Kissinger writes that Israel wanted to use "hysteria or blackmail" to convince the US to resupply it, and one way to do that was by threatening to use nuclear weapons if conventional ones ran out, at which point Israeli leaders would've felt that all was lost. The US resupply actually dwarfed the Soviet one, and was actually ordered shortly after the Soviet one, on October 9. But there was a delay, which some blame the Pentagon for and others blame the administration for (playing a "double game", some Israeli leaders argued), and the US airlift began to land on October 14, by which point the tide had already turned in favor of the Israelis, but the munitions helped the Israelis keep the momentum without fear of supply shortages, and the airlift ended up dwarfing the Soviet airlift.

The Israelis didn't outright threaten to use the weapons, either, according to some authors. Rather, they made an easily detectable move to arm their Jericho missiles, which could be fitted with nuclear warheads, and placed them on high alert.

However, it does appear that this was more of an empty threat in reality. According to Israeli historian Avner Cohen, when Israeli Defense Minister Moshe Dayan ordered to prepare Israeli nuclear weapons for a defensive "demonstration", he was overruled by the Israeli Prime Minister, Golda Meir, according to an interview Cohen conducted with a former senior advisor in the Israeli government Azarayahu "Sini" Arnan, who spoke with Cohen just 10 months before he passed away. Dayan advocated to demonstrate, and not to target enemy forces with the weapon itself, to try and buy time and scare the invading forces. It's unclear what he meant by this, but we can speculate that he meant either an underground test (less likely, due to the preparation involved) or an aboveground airburst in an unpopulated area, likely over Egypt or Syria (to show what Israel could do). As mentioned, this was overruled by Meir.

This information has to be taken with a grain of salt. "Sini" was over 90 years old at the time, and memory is imperfect. Nevertheless, this information provides some interesting evidence that leans in the direction most historians suspect: that Israel was using the nuclear threat coercively, and would have had a very tough decision if it ran out of conventional arms but had not yet made that decision, despite showing and preparing apparent nuclear launches.

So no, Golda Meir did not threaten to use nuclear weapons on Damascus. Rather, she used a very interesting, vague tactic to signal nuclear readiness and scare others into thinking she might use nuclear weapons on Damascus. This did not have to do with entering "Israel proper" necessarily, and had far more to do with American resupply efforts, and possibly giving pause to invading Arab forces.

Most of our understanding of this comes from off-the-record sources, however. We don't have a very thorough understanding of the full extent here, because this is highly classified both in US records and in Israeli ones. In fact, it was very surprising that the interview with "Sini" slipped past Israel's military censors, who watch for the revealing of classified information in Israeli media.