Firstly, you can go from nothing to full on labour pretty quickly so not having any signs at 39 weeks isn't anything to worry about. Most people are 'overdue' with their first (I am assuming it's your first) and bear in mind that 40 weeks is a NHS guideline - other countries think 41 weeks is normal!
Secondly I was exactly the same as you, I really wanted a natural birth, I had a pool home birth planned. I went overdue, I had a breech baby then a back to back baby, both of which tend to extend gestation time. If you are concerned get them to double check the lie of your baby - ask how engaged the head is. If it's not engaged push to make sure they are sure the baby is in the correct position. I say this as I only found out at 41 weeks that my baby was breech - no one thought it remotely odd that until that point that my baby hadn't engaged at all and were all happily convinced she was head down.
Anyway, without those complications I think I'd have gone into labour naturally. But I didn't, ended up being induced and it wasn't all I'd hoped for. But we did as much as we could to make the experience like I wanted (music, low light etc on the labour ward) and at the end of the day once your sungly newborn is here, it won't matter how they arrived!
Also if they do talk about induction, you can push it back, I waited as long as possible before starting mine. I ended up having my DD at 40+15. Don't forget that at 40 weeks they offer sweeps and that will give you an indication of how ready your cervix is etc.