I move my unit into a BF controlled by my opponent. Can I play Flurry of Feathers to put 4 birds on the BF I am attacking?

No, you cannot play the Bird tokens from Flurry of Feathers onto the battlefield you are attacking.

According to RiftJudge FAQ #10013, you cannot play units to a battlefield you do not control. When you are attacking a battlefield, it is under your opponent's control, and units must be played to your base or a battlefield you control.

Why this is the case:

1. Location Control: When you move a unit into a battlefield controlled by your opponent, you do not gain control of that location; it becomes "Contested." Because you do not control it, you cannot legally play units there.
2. Token Rules: As established in the FAQ, token units follow all normal rules for playing units, which strictly require you to play them to your base or to a battlefield you already control.

While Flurry of Feathers has the [Reaction] tag—meaning you could play it while the showdown chain is closed—you are still restricted by the location requirements for playing units. If you play Flurry of Feathers, you must choose a valid location for the tokens (your base or a battlefield you control) rather than the one you are attacking.