"Wee" is a Scottish word for "Small" or "little". It has nothing to do with peeing! If a Scotsman saw a small boy he might say "Och, there's a wee lad". A "wee trip" is therefore a short journey that you expect to return from soon (but longer than the seconds you were implying!)
I don't think it's normally used in Australia, but I'm sure many Australians have heard of it and may use it in jest. Or if they were originally Scottish of course!