Not automatically, no… - that’s what the built in Unique IDs are for (also Slugs are checked automatically for uniqueness, but if they are not then a number is appended to them, so probably not what you want in this case).
Could you just create a field called ‘booking number’, then whenever the booking is created, you cold perform a search in your database for the count of all the existing bookings.
Once the count has been performed, you could +1 to that number - giving it a sequential booking number.
I would usually go with something simple like current date/time formatted as yymmddhhmm. You can break it to something like yymmdd-hhmm it looks a little nicer.
I think this is the way to go since the new WU system will make this kind of searching and comparing extremely expensive and therefore not worth the struggle.
Current timestamp plus a little salt can do the job. Depends on how many users you have and if collusions are likely or not.