Yes, God does incarnate. Hindus believe in incarnation (called avatars or descent). The incarnation of a deity in human or animal form is to counteract some particular evil in the world to establish balance.
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यदा यदा हि धर्मस्य ग्लानिर्भव- ति भारत ।
अभ्युत्थान- मधर्मस्य तदात्मानं सृजाम्यहम्- ॥४-७॥
परित्राणाय- साधूनां विनाशाय च दुष्कृताम्- ।
धर्मसंस्था- पनार्थाय सम्भवामि युगे युगे ॥४-८॥
Lord Krishna in Bhagavad Gita tells his best friend Arjun (chapter 4.17):
Yada yada hi dharmasya glanirbhavati bharata
Abhythanamadharmasya tadatmanam srijamyaham
Paritranaya sadhunang vinashay cha dushkritam
Dharmasangsthapanarthay sambhabami yuge yuge
Whenever there is a decline of righteousness [dharma] and rise of unrighteousness then I send forth Myself. For the protection of the good, for the destruction of the wicked, and for the establishment of righteousness, I come into being from age to age (I descend myself).
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Lord Vishnu (lord of sustenance or protector of humanity) had 10 popular incarnations (or appearances): Matsya (fish), Kurma (tortoise), Varaha (boar), Narasimha (half man, half lion), Vamana (dwarf), Parashurama (Rama with the axe), Rama (hero of the Ramayana epic, in Treta yuga/time period), Krishna (author of Bhagwad Gita, in Dvapar yuga/time period), Buddha (established Buddhism), and Kalki (the incarnation yet to come). The number of Vishnu’s avatars is sometimes extended or their identities changed, according to local preferences.
Actually, we are all God incarnate. Only in our case, we don’t remember our essential divinity.
Other reading material: https://www.britannica.com/topic/avatar-Hinduism
Relevant videos: Swami Nikhil Anand Hinduism Q & A: Does God Incarnate
Contributor: Nilesh Chaudhary