We formulate and solve the fixed horizon linear quadratic covariance steering problem in continuous time with a terminal cost measured in Hilbert-Schmidt (i.e., Frobenius) norm error between the desired and the controlled terminal covariances. For this problem, the necessary conditions of optimality become a coupled matrix ODE two-point boundary value problem. To solve this system of equations, we design a matricial recursive algorithm and prove its convergence. The proposed algorithm and its analysis make use of the linear fractional transforms parameterized by the state transition matrix of the associated Hamiltonian matrix. To illustrate the results, we provide two numerical examples: one with a two dimensional and another with a six dimensional state space.