I don't think it's possible but want to be sure and the exact reason.
I don't think it is possible because the atmosphere on the moon is less than one hundred trillionth ($10^{−14} $) of Earth's atmospheric density at sea level. So the moons' air pressure would also be very (equally) small. I suppose that somenone's power to suck an almost 'vacuum' to create an underpressure in the straw is not possible. Perhaps to increase the surface of your glass could help a bit.
I think that gravity doesn't have to do with it a lot because that is the same for the liquid in the straw and outside on the surface of the glass.
Perhaps I forgot something.... Because of big variations of temperature on the moon I suppose the liquid is not frozen nor evaporated. And probably you should protect yourself and provide yourself with oxygen etc, but they are left out for simplicity of the problem.