  We examine the enumeration of certain Motzkin objects according to the
numbers of crossings and nestings. With respect to continued fractions, we
compute and express the distributions of the statistics of the numbers of
crossings and nestings over three sets, namely the set of $4321$-avoiding
involutions, the set of $3412$-avoiding involutions, and the set of
$(321,3\bar{1}42)$-avoiding permutations. To get our results, we exploit the
bijection of Biane restricted to the sets of $4321$- and $3412$-avoiding
involutions which was characterized by Barnabei et al.~ and the bijection
between $(321,3\bar{1}42)$-avoiding permutations and Motzkin paths, presented
by Chen et al.~. Furthermore, we manipulate the obtained continued fractions to
get the recursion formulas for the polynomial distributions of crossings and
nestings, and it follows that the results involve two new $q$-Motzkin numbers.
