Yes, that is a straightforward algorithm.
There is no special mathematical or computational meaning that is attached to the word "straightforward" here. It just means what it means in plain English, as by Google search by me today,
uncomplicated and easy to do or understand.
To be slightly pedantic, I would avoid saying "a straightforward algorithm refers to the algorithm which intuitively solves a specific problem". Usually, we use intuition when we rely on some instinct, heuristics or experience . The algorithm here, which computes the square of a matrix, is implemented strictly according to the definition of matrix multiplication. This implementation is completely mechanical, assuming the ability to transform a specification of a procedure into program.
"Straightforward" is, indeed, the most appropriate word to describe the algorithm.