Double cosets of compact quantum groups with respect to their subgroups are considered and cases of a Gel'fand pair and a strict Gel'fand pair are distinguished. It is shown that every strict Gel'fand pair of compact quantum groups generates a normal commutative hypercomplex system with a compact basis and a commutative discrete hypergroup which are in duality to each other. The examples of strict Gel'fand pairs of compact quantum groups are considered and characters of the corresponding hypergroups are described in terms of q- orthogonal polynomials.