If you are making a 5 gallon batch in a 5 gallon carboy, yes that will cause issues. If it's a smaller batch, probably not. You need headspace for the yeast to rise into (krausen) or you will have quite a mess. I would recommend you rack off about 1/2 a gallon into another container and use a length of tubing and a small container of star San, vodka or just water as a "blowoff tube". This time tomorrow, fermentation will likely be pretty vigorous and will likely shoot right out of the typical beginner airlock if there isn't enough headspace.
Be very careful with the glass. Serious injuries come from broken glass carboys. Plastic or stainless are much safer. Search this forum for a compendium of reasons to avoid glass