Not to any significant extent, especially if the solvent is pure ethylene glycol.
Sodium bicarbonate has a pKa of about 6.5, and EG has a pKa of about 15, so in an aqueous solution containing equal concentrations of NaHCO3 and EG, you would expect that fewer than 1 in 100000000 EG molecules would be deprotonated.
The autoprotolysis of anhydrous pure ethylene glycol can be expected to play a nearly neglectable role, but that contribution is believed to increase noticeably with the water content. Under favourable conditions, it appears that initially anhydrous (carefully dried) ethylene glycol may possibly be able to react with (dry) dissolved sodium bicarbonate, to some (limited) extent, accordingly: NaHCO3 + (CH2OH)2 ↔ CH2OH―CH2ONa + H2O + CO2, what should be inhibited by either dissolved CO2 and H2O. If indeed formed, the organic salt monosodium ethyleneglycolate (also named sodium glycoxide, or sodium 2-hydroxyethanolate) should partially dissociate itself: CH2OH―CH2ONa ↔ CH2OH―CH2O- + Na+. Finally, even traces of water could be then expected to react with the ethyleneglycolate anion, regenerating ethylene glycol: H2O + CH2OH―CH2O- ↔ (CH2OH)2 + OH-.