Given the orbit of the Earth around the Sun and the position of the Sun in the Galaxy, in which season are we (the Earth) closer to the galactic center?
The radio source Sgr A* is at RA 17h46m Dec -29°, or about 6° south of the ecliptic at longitude 267°. The Sun appears most nearly opposite that position, ecliptic longitude 87°, about three days before the June solstice. At that time of year, the brightest part of the Milky Way is highest in the sky around local solar midnight.
However, the distance from here to Sgr A* is about 7.9 kpc or 1.6 billion au. The Earth is only infinitesimally closer to the galactic center in June than in December.