If you could find the quote, I could be certain.

There was no referendum I'm aware of, and if there was a referendum the idea that it would've gone in favor of the heavily minority Jews establishing a state is actually quite unlikely. Especially a referendum that would've been conducted mid-war, with an Arab revolt going on under the auspices of establishing an Arab state to replace the Ottoman empire.

What they are likely referring to is the 1917 Balfour Declaration, and mistaking it for some referendum. What it said was that the British were in favor of establishing a "Jewish National Home" in Palestine. This was no referendum, it was not representative of what the majority in Palestine wanted (which was unclear, though at the time it appeared a pan-Arab state or continuing to live under Ottoman rule were the major wants, with some smaller groups in favor of full independence from any larger state), and if it were truly a referendum then the Balfour Declaration would not have been met with such ire and led to (as an example) the 1920 Nebi Musa riots, or the subsequent riots in 1921.