Apply Outer to list of matrices and vectors

I believe what you want is

Outer[#2.#1.#2 &, M, V, 1]

First of all, you don't need to Transpose vectors in Mathematica. Dot knows what to do automatically. Second, you need to use the optional extra argument for Outer that tells it the level at which the "object" that you are feeding to the function lives. Otherwise, Outer seeks the lowest level by default. Since your "objects" are at level 1 in both lists (the elements of the lists which are either matrices or vectors), use the level-spec 1 as the extra argument to Outer.

Just use Table

SeedRandom@12;
m = Table[RandomInteger[5, {2, 2}], 3];
MatrixForm /@ m

m=$\{\left( \begin{array}{cc} 1 & 1 \\ 0 & 0 \\ \end{array} \right),\left( \begin{array}{cc} 5 & 4 \\ 1 & 4 \\ \end{array} \right),\left( \begin{array}{cc} 5 & 3 \\ 0 & 5 \\ \end{array} \right)\}$

v = Table[RandomInteger[5, 2], 3]

v={{3, 4}, {1, 1}, {2, 5}}

Table[v[[j]].m[[i]].v[[j]], {i, Length@m}, {j, Length@m}]

{{21, 2, 14}, {169, 14, 170}, {161, 13, 175}}

Second Solution

#2.#1.#2 & @@@ Tuples[{m, v}] // Partition[#, 3] &