If a player attempts to lay an egg but has no available nest icons on their player mat or dragons, they may not lay the egg. The rulebook explicitly states:

"If you do not have an empty nest icon available, then you may not lay an egg." (Page 8)

This means the action fails, and the player cannot place the egg.

Source: Wyrmspan Rulebook (pp. 14, 2, 6, 8)