Does Bold fighter allow Quads?

Bold Fighter doesn’t Quad on its own. H!Kagero has 21 Spd with Bold Fighter and initiates combat with this Green Cavalier that has 3 Spd, She only Doubles him.

No how it usually works is in the wording.

"Unit attacks twice" is the key phrase for quad. "Follow up" is the regular effect of when you outspeed people.

Teach a man to fish

If you can double the ennemy unit and have bold fighter, you'll double against an ennemy who have an ability that deny double like L! Hector or wary fighter.

But only with brave weapon or double lion you can quad. Because brave weapon make the user attack TWICE, when with a non-brave weapon, you can only double (wich is another mecanic)