Do Hindu Gods too seek Moksha?

The concept of God is very multifaceted in Hinduism. At the grand perspective, we have Brahman (or Parabrahman) — the only existing, all pervading God beyond name and form.

Then we have functional God forms (Brahma, Vishnu, and Shiva) who are verily the manifestation of Brahman considered with separate identity for the tasks of Creation, Protection and Destruction. They are ever existent.

Perspective 1: Such a functional God is beyond boundaries and limitations. He is ever perfect. He is nitya mukta – ever free.

Perspective 2:  

Then we have the concept of Avatar — God descending to earth in human form.

When God comes down to earth as an Avatara Purusha He may sport a lila (play) of doing tapas (undergoing austerities) and attaining liberation as a human being, before starting teaching others. Some avatars may even sport a lila of taking instructions from Gurus too. E.g. Rama learning Yoga Vashistha from Rishi Vashishta; Krishna learning Kundalini Yoga from Rishi Sandipani; Sri Ramakrishna Paramahamsa learning Nirvikalpa Samadhi from Totapuri.

Perspective 3: 

Then we have the concept of God residing as in-dweller in each one of us.

God as an indweller in all keeps himself totally hidden on account of the ego of the individuals. When an individual, getting Viveka & Vairagya (Discrimination & dispassion) starts seeking the truth about his true status, he starts yearning for Moksha. “At that point of time God appears as the external Guru and and turns the mind of the seeker inwards; God the indweller pulls him in from inside” (as stated by Ramana Maharshi)