I don't know based on the rulebook.

The provided rulebook context explains that some bird powers, such as the Western Meadowlark's, allow "all players" to gain a resource (e.g., "All players lay 1 [egg] on any 1 bird. You may lay 1 on 1 additional bird." — page 8). However, the rulebook does not specify whether other players may activate their own bird powers in response to gaining a resource from another player's bird power.

The context does explain how ONCE BETWEEN TURNS (pink) powers work: "These powers may be triggered on opponents’ turns. You can only use a pink power once between each of your own turns (if an opponent triggers it)." (page 10). But it does not clarify if brown powers or other types of powers can be triggered in response to resources gained from another player's bird power.

To answer this question definitively, the rulebook would need to specify:

• Whether gaining a resource due to another player's bird power counts as "using the corresponding habitat" or otherwise triggers powers.
• If and how players may activate their own powers in response to such effects.

Source: pages 8, 10.

Source: Core Rulebook (pp. 1, 10, 7, 8)