In the present paper, by using of the coefficients of the characteristic polynomial of matrix B and the so-called Leverrier algorithm, the explicit solutions to the Sylvester-conjugate matrix equation AX – XB = C (including the Lyapunov-conjugate matrix equation as special case) have been constructed. While one of the explicit solutions is stated as a polynomial of coefficient matrices of the matrix equation, one of the explicit solutions is expressed by the symmetric operator matrix, controllability matrix and observability matrix. Comparing to the existing results, there is no requirement on the coefficient matrices. At the end of this paper, one numerical example is shown to illustrate the effectiveness of the proposed method.