Time complexity for FPT algorithm
Suppose that you have a graph and each vertex is colored with one of three colors, and let the set of different-colored vertices be $A, B, C$.
Now, try to find a triangle with one vertex from each partition.  For each vertex $a \in A$, try let $B_a$ and $C_a$ be $a$'s neighborhood in $B$ and $C$, respectively.  Now you only need to check if there is an edge from $B_a$ to $C_a$.  If there is one, you have your triangle, otherwise, you try another vertex from $A$.
Let us call this subroutine colorful_triangle(G, A, B, C).  Observe that it is not necessary that $A \cup B \cup C = V(G)$.

Let $G$ be the graph, $\chi$ the coloring and  $\pi$ a permutation of colors.
Let $\chi(i)$ be the set of vertices that received color $i$.
Now let $A = \chi(\pi(1))$, $B = \chi(\pi(2))$, $C = \chi(\pi(3))$.  Run the subrouting colorful_triangle(G, A, B, C).  If successful, continue with
$A = \chi(\pi(4))$, $B = \chi(\pi(5))$, $C = \chi(\pi(6))$, and so on.
If you at some point are unsuccessful, move on to the next permutation.