Seeing as griffins are much bigger creatures than birds, plenty more twigs and sticks would be required and so larger size too. So how would griffins go about collecting materials for their nest? One of the ways I have pictured is a griffin flying up to some tree and stripping it of its branches, while possibly claiming a small rodent as a meal at the same time :).
These particular griffins have the head, wings and front talons of an eagle and the back legs and tail of a lion. They like high places such as mountains, abandoned towers, castles etc to set their nests upon and to watch over the surrounding lands. They do lay eggs which is why nests are necessary and finally valuable small treasures or item they have can be hidden underneath the nest for safe keeping.