You can both stand for election to the EC but only if you are nominated by other owners. There are various quirks in this part of strata law – this is what it says:
Schedule 3: Part 2 Constitution of executive committees for strata schemes with more than 2 lots
(5) A person who is co-owner of a lot may not be a candidate for election as a member of the executive committee unless the person is nominated for office:
(a) by an owner who is not a co-owner of the lot, or
(b) by a co-owner of the lot who is not a candidate for election as a member.
(6) An owner of a lot who is not a co-owner of the lot may nominate himself or herself for election as a member of the executive committee.
My interpretation of that is that if you both want to be on the EC you both have to be nominated by other owners. The reason for that is 5 (b) that states you can't be nominated as a candidate by your co-owner if that person is standing for election themselves.
So find two other owners who want you to both to be on the EC and you'll be fine. But you definitely can't nominate each other for two vacancies.