No, because our calendar is tied to the tropical year (the mean time between one solstice and the next), rather than the sidereal yesr (the time it takes the Earth to return to the same position in its orbit). So the equinoxes will keep the same position in the calendar (averaged over time). What will change is the apparent position of the sun against the background constellations, at the time of equinoxes and solstices.
For instance, the northern spring equinox will stay in late March, but the sun's position at equinox will gradually shift from Pisces to Aquarius.

Grant Hutchison