Imagine that the cells of the 2x2 grid are denoted in this way

A1 A2

B1 B2

There are 2 basic cases: (a) A2 and B1 are of different colors (and different from A1 and B2),

and

(b) A2 and B1 are of the same colors (and different from A1 and B2)

Consider case (a) first.

(a) For A1, I can use any of 6 colors;

then for A2 I can use any of remaining 5 colors;

for B1 I can use any of remaining 4 colors;

for B2 I can use (and I must use) one of remaining 3 colors OR the same color as A1.

In all, I have 6*5*4*(3+1) = 6*5*4*4 = 480 different colorings in case (a).

Consider case (b) next.

(b) For A1, I can use any of 6 colors;

then for A2 I can use any of remaining 5 colors;

for B1 I use the same color as A2, so I have no choice;

for B2 I can use (and I must use) one of remaining 4 colors OR the same color as A1.

In all, I have 6*5*1*(4+1) = 6*5*1*5 = 150 different colorings in case (b).

Cases (a) and (b) are DISJOINT, so the ANSWER to the problems' question is this sum 480 + 150 = 630.