Exact Solutions with Two Parameters for an Ultradiscrete Painlev\'e Equation of Type $A_6^{(1)}$

An ultradiscrete system corresponding to the $q$-Painlev'e equation of type $A_6^{(1)}$, which is a $q$-difference analogue of the second Painlev'e equation, is proposed. Exact solutions with two parameters are constructed for the ultradiscrete system.