Not as such. I don't know why they described it in such terms, but usually grapple is its own action, not an unarmed attack. An amulet of might fists, for example, should not increase your grapple CMB unless I've missed some rule somewhere.
Barring FAQs and erratas, I think this is a tricky question.
Add any bonuses you currently have on attack rolls due to spells, feats, and other effects. These bonuses must be applicable to the weapon or attack used to perform the maneuver.
I don't know what the RAI is. It appears that if you trip with any weapon you can add the enchantment bonus. But if blahpers is correct, then even the more expensive Amulet of Mighty Fists won't add to your attack roll.
Even though you don't normally make grapple checks with a weapon, there are weapons with the grapple ability that do give you bonuses to attack with them which would mean they have to give you a bonus for the enchantment that goes along with it.
In fact the errata for trip and disarm specifically state that you CAN add the enchantment bonus for any weapon you use. This includes the bonus from Unarmed Strikes, which would allow you to gain the benefits from the Amulet for at least those two maneuvers.
Here is where it gets really tricky with RAI vs. RAW. If you have a Dan Bong (light, one-handed weapon +2 to grapple checks) you would get a total -2 to grapple checks because:
Humanoid creatures without two free hands attempting to grapple a foe take a –4 penalty on the combat maneuver roll.
And if you had two Dan Bongs, I guess you would have a +0? or can you not use both even though you need both hands to make the check? But if you don't automatically disregard the -4 penalty above, why would you ever use the weapon? It is only a d3 to damage. It would be a serious corner case to gain any real benefit from it.
So clearly you can use a weapon to grapple. There are 3 weapons in ultimate combat that have the grapple quality.
If you follow the same logic applied to trip and disarm (that you can use any weapon if you can use one weapon) then you can use Unarmed Strikes for grapple attempts. Thereby gaining the benefits of the Amulet of Mighty Fists.
I have no idea if this is RAW or RAI or TS*. But I don't think it is much of a leap of logic to come to this conclusion. Since almost every Combat Maneuver can add a weapon's bonus, why do we assume grapple can't?
Just typing out loud, don't kill me.
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*Totally Stupid
Agreed on the trickiness. I'm no longer certain what the case is here.
Hmm, what about dirty trick...?