Extracting faces and vertices from a mesh

Hello guyz, I am doing some plane stress FEA using Matlab and need the matrix of the quadrilateral elements (faces) and their corresponding nodes (vertices) to input into matlab file. I have a square matrix whose elements are numbers which could be 1 or other numbers. E.g: A = [1 1 1 1; 1 1 1 1; 1 2 3 1; 1 2 4 1]. If the cells with value of 1s are meshed, the resulting geometry is what i need (image is attached). An element is 1 unit in length and width. The challenge I am facing is how to extract the vertices and faces into a matrix form (shown in the attached file) using matlab. Please can anyone assist me. I would be very happy because this is my MSc thesis.

Olu