Hmm, so picking a 175ml Martini glass - it weighs 185g. If we assume near half of that is in the stem and base, lets call it 100g of glass in near contact with the cocktail proper.
Assume Borosilicate glass, even though it may not survive the thermal shock, we can postulate similar glasses that can, and we might assume they will have similar thermal properties.
(Borosilicate glass will survive a 165 K differential - which is not enough to manage liquid nitrogen at its boiling point to room temperature, 77K to say 300K.)
Specific heat borosilicate glass is pretty well 10[sup]3[/sup] Jkg[sup]-1[/sup]K[sup]-1[/sup]
So our 0.1 kg of glass will hold about 77 * .1 x 10[sup]3[/sup] = 7,700J when dropped in LN[sub]2[/sub]
Assume a Martini. Gin is between 40 and 50% alcohol by volume. Vermouth is say 18% ABV. Nobody will agree on the perfect ratio - so lets just go with 1oz Vermouth, 4oz Gin.
assume the final mix is about 40% ABV, and has a volume of 5oz = 150ml.
Ethanol has a specific heat of: 2.4 x 10[sup]3[/sup] Jkg[sup]-1[/sup]K[sup]-1[/sup]
Water is: 4.2 x 10[sup]3[/sup] Jkg[sup]-1[/sup]K[sup]-1[/sup]
So the 40:60 mix (assuming all sort of things here) is: about 3.5 x 10[sup]3[/sup] Jkg[sup]-1[/sup]K[sup]-1[/sup]
Density of a 40% ABV mix is 0.94. Close enough to 1 not to matter.
Our Martini at room temperature (call it 300K again) holds .15 * 3.5 * 10[sup]3[/sup] * 300 = 160,000J About 20 times the energy of the cooled glass.
This ratio is so large (plus the much higher specific heat of the Martini versus the glass) that there is no point calculating the final energy of the combined system and working out the temperature. As a good enough approximation assume the glass sucks about 1/20[sup]th[/sup] the energy out of the Martini. It will drop at most about 15K or 27F. (In reality a little less, but close.)
That isn’t startling, nor is it trivial. But keeping your Gin in the freezer probably wins.