It depends on how well your compound will run on silica, but yes, Et3N is very stationary in flash chromatography.
However, it's about 100x easier to just wash your product with 1M HCl. I'm assuming your product doesn't have free amines on it, or else you'll have difficulty running that on a flash column. But if it doesn't, then just take it up in solvent (ether or EtOAc should be fine) and wash it a couple times with mild HCl, I do that all the time to get rid of amine bases from my crudes.