The quote is false.
Here is what Meir said, referring to when the bulk of Zionists returned to their land in the first half of the 20th century, quoted in the June 15, 1969 Sunday Times:
There were no such thing as Palestinians. When was there an independent Palestinian people with a Palestinian state? It was either southern Syria before the First World War, and then it was a Palestine including Jordan. It was not as though there was a Palestinian people in Palestine considering itself as a Palestinian people and we came and threw them out and took their country away from them. They did not exist.Notice that she stated it in past tense, and given the timeframe that she was referring to, her statement is indisputable. (The correct statement is quoted much less frequently on the Internet.)
There is another interesting thing about the timing of her statement.
The first time that Palestinian Arabs were recognized as a "people" by the UN was not in the 1940s or 50s. It was in December, 1969, six months after Meir spoke on the topic. In GA Resolution 2628, it refers to the "people of Palestine" - the first time that this term was used by the UN to refer exclusively to Palestinian Arabs.
And the first time that the term "Palestinians" was used by the UN in reference to Palestinian Arabs was a year later, in A/RES/2628 in November, 1970.
Isn't it interesting that for over twenty years after the Palestinian Arab exodus, the UN never referred to them as a people? Only after the Six Day War, when the Arab world changed its tactics from bragging about "throwing the Jews into the sea" and into "Jews as tyrants, Arabs as victims" did the UN pick up on this new construct of recognizing a people and a state whose existence were denied even by Arabs only a few years beforehand.
Not to mention afterwards, as this (Hebrew) video shows, as Palestinian Christian politician Azmi Bishara says explicitly:
Or the famous quote from the PLO's Zuheir Mohsen denying the existence of a Palestinian people in 1977.
(I did not research the UN resolutions exhaustively; I came upon this in a footnote in this paper. H/t to Biodegradable and Jed for post-1970 quotes.)