The Moor-Penrose generalized inverses (M-P inverses for short) of matrices over a finite field Fq2, which is a generalization of the Moor-Penrose generalized inverses over the complex field, are studied in the present paper. Some necessary and sufficient conditions for an m×n matrix A over Fq2 having an M-P inverse are obtained, which make clear the set of m×n matrices over Fq2 having M-P inverses and reduce the problem of constructing and enumerating the M-P invertible matrices to that of constructing and enumerating the non-isotropic subspaces with respect to the unitary group. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing the results in geometry of unitary groups over finite fields.