  Recently, Merino extended the classical relation between the $2n$-th
Fibonacci number and the number of spanning trees of the $n$-fan graph to
ribbon graphs, and established a relation between the $n$-associated Mersenne
number and the number of quasi-trees of the $n$-wheel ribbon graph. Moreover,
Merino posed a problem of finding the Lucas numbers as the number of spanning
quasi-trees of a family of ribbon graphs. In this paper, we solve the problem
and give the Matrix-Quasi-tree Theorem for a bouquet with exactly one
non-orientable loop. Furthermore, this theorem is used to verify that the
number of quasi-trees of some classes of bouquets is closely related to the
Fibonacci and Lucas numbers. We also give alternative proofs of the number of
quasi-trees of these bouquets by using the deletion-contraction relations of
ribbon graphs.
