An analytic formula is reported for general matrix elements of the microscopic quadrupole-quadrupole operator in the U(3)-boson approximation. The complete infinite-dimensional basis of A-fermion wave functions is compatible with the harmonic-oscillator shell model and consists of np-nh configurations, with spurious center-of-mass excitations removed, which are symmetry adapted to the Elliott U(3) and symplectic Sp(3,R) models. The formula expresses the general ${\mathit{Q}}_{2}$\ensuremath{\cdot}${\mathit{Q}}_{2}$ matrix element with respect to this complete orthonormal basis as a Racah SU(3) U coefficient times a closed-shell matrix element. An oscillator closed-shell matrix element of ${\mathit{Q}}_{2}$\ensuremath{\cdot}${\mathit{Q}}_{2}$ is a square root of a rational function of the integer quantum numbers of the U(3) basis.