Can we have a poly time reduction from 2-SAT to 2-Coloring problem?
As rotia mentions in their comment, to make the question meaningful we must restrict the power of the allowable reductions. The two most obvious choices are to allow only logspace reductions or only AC0 reductions.

Since 2SAT is NL-complete (with respect to logspace reductions) whereas 2COL (2-colorability) is in L (this is a non-trivial result which follows from Reingold's theorem), if there is a logspace reduction from 2SAT to 2COL then L=NL, which is considered unlikely. Hence we do not expect such a reduction to exist; a fortiori we do not expect an AC0 reduction to exist.