How many 4 vertices connective graphs not including a triangle? How many 4 vertices connective graphs not including a triangle? 
I am thinking the answer might be 2, for one is a square, another is straight. But I am not sure.

I assume that you mean simple graphs, i.e., graphs with no loops or multiple edges. Every tree on $4$ vertices satisfies your condition, because a tree has no cycles at all, and you’ve missed the tree with a vertex of degree $3$. If the graph is not a tree, it must contain a cycle, that cycle must be a $4$-cycle, and it’s easy to check that adding any edge to a $4$-cycle creates a triangle. Thus, the graphs are the $4$-cycle and the trees on $4$ vertices.
