I have a matrix of decision variables in X (size mxn) and would like to use the expression x*V*x’ in one of my cone constraints (with V positive definite and size n*n). Unfortunately cvx complains that this is not a scalar quadratic form. What is the best way to use these kind of matrices as a function of decision variables in the constraints?

The construction x * V * x’ , where x is a matrix, is neither convex nor concave. CVX simply cannot use it. Unless you can rewrite the larger constraint in a compatible fashion, perhaps using an LMI, you are not going to be able to specify this problem in CVX.