Hi we recently took under our wing a so we are told 6 year old toy bichon frise adorable girl very good apart from the barking she barks at nothing at times as soon as shes let out she starts anyone walking past, bikes. Etc it is quite an issue and need some advice on trying to stop it? Small dog stndrome is known and i was advised to hold a treat to her nose and as she quietens to sniff it say the word quiet and feed the treat? But then im concerned she will just bark for the sake of getting the treat for being told to be quiet?

As with all training involving a reward, first you reward for every compliance, then for the fastest and then gradually phase out to random treating.

What your trying to do is get her to offer and alternative behaviour incompatible with what she is doing that you wish to curb.

Your positive attention has to be better than what she is doing.  Eventually your praise and or affection will be reward enough.

Dogs often bark when they want something, such as going outside, playing, or getting a treat. Separation Anxiety/Compulsive Barking, Dogs with separation anxiety often bark excessively when left alone.Compulsive barkers seem to bark just to hear the sound of their voices.

Timing is important with the method you describe, as is increasing duration.  So initially, you would say 'quiet', produce the treat right in front of her nose, and hold it there for 4 or 5 seconds before praising and rewarding her.  This way you've got a decent bit of non-barking that's getting the reward.  This can still fall foul of her learning to bark then shut up to get the treat, though, so once she's got that bit down, you would start waiting a second longer each time so that she learns that it's going quiet and staying quiet that gets the treat.

All of that said; you need to address the reason for the barking, which is often either overexcitement or nerves (more often the latter IME).  The remedy is pretty similar either way - give her space so she's calmer (proximity increases stress increases reactivity, regardless of the reason for it), then reward calm behaviour.  And don't just wait for it to happen, encourage it!  Try keeping her attention with food and rewarding quickly if she sees a trigger but doesn't react (as in instantly reward).  Talk to her, move her further away if need be, engage with her to help her stay calmer and give you more to reward.

That's a greatly described and nicely presented.
I will also learn from your message. :)