I think the answer is
5
using the following coloring:
For other board sizes,
5 is sufficient as well; the pattern can just be repeated. (Of course, a 2x2 board needs only 4 colors because there are only 4 squares. And does 1x1 even count as a board?)
Reasoning:
Consider a square not on the edge of the board with its 4 orthogonal neighbours; they all have to have different colors since each pair is part of a tromino. Therefore, we need at least 5 different colors; the pattern shows 5 is sufficient.