Not likely.
Part 1 (Me Complaining)
If we look at Fast Healing, we see as you noted:
The creature regains hit points swiftly, usually 1 or more per round, as given in the creature’s entry. Except as noted below, fast healing is just like natural healing. Fast healing does not restore hit points lost from starvation, suffocation, or thirst, nor does it allow a creature to regrow lost body parts. Unless otherwise stated, it does not allow lost body parts to be reattached. Fast healing continues to function (even at negative hit points) until a creature dies, at which point the effects of fast healing end immediately.
At which point the natural inclination is to look at the rules for Natural Healing:
Natural Healing: With a full night's rest (8 hours of sleep or more), you recover 1 hit point per character level. Any significant interruption during your rest prevents you from healing that night.
Nothing about fast healing seems to change the need to rest. This means the character needs to be sleeping/resting in order to gain the benefits. Note, that it can be argued that fast healing does change the required timeframe for the rest though, from 8 hours to 1 round. This would mean that a creature would only need to rest for a round in order to gain the healing from fast healing.
Part 2 (Getting To The Point)
Now, it's silly to think that fast healing requires a creature to go ahead and sleep in combat in order to take advantage of it. I've never seen it run that way, nor have I even considered that could be a take before reading these rules today. And while it kind of makes some sense, fast healing speeding up the natural healing process, but still requiring sleep to heal, I don't think that's how I'd recommend running fast healing. Which leads me to my next point, read the ability.
If we look at Vivacious, we see:
Some gnomes retain a trace of the vitality of the First World. These gnomes recover 50% more hit points (minimum 1) whenever they recover hit points from rest. Whenever they are healed of hit point damage by a spell, they heal an additional amount equal to 1/2 the spell’s caster level (minimum 0). The extra healing does not apply to spells that grant fast healing or similar effects. This racial trait replaces gnome magic and keen senses.
See, there's a qualifier there, the ability only applies to hit points gained from resting, not hit points gained from natural healing (which just happens to require rest). While fast healing functions like natural healing, the way I've seen it run, it does not require rest to heal. As such, it's not hit points recovered from rest.
TLDR; (The Finale)
Vivacious does not apply to the healing from fast healing, unless your game makes fast healing require you to be resting for it to apply.
