I think the answer is

>! 5

using the following coloring:

>! [![enter image description here][1]][1] 

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 triomino. Therefore, we need at least 5 different colors; the pattern shows 5 is sufficient.


  [1]: https://i.sstatic.net/8xLrN.png