Which God or Goddess is known as God of Silence in Hindu Religion?
Dakshinamurthy, an aspect of Shiva and deity of wisdom, can be regarded as the god of silence since he parts knowledge by observing silence:
मौनव्याख्या प्रकटित परब्रह्मतत्त्वं युवानं वर्षिष्ठांते वसद् ऋषिगणैः आवृतं ब्रह्मनिष्ठैः । आचार्येन्द्रं करकलित चिन्मुद्रमानंदमूर्तिं स्वात्मारामं मुदितवदनं दक्षिणामूर्तिमीडे ॥१॥
English Translation: Whose exposition through profound silence awakens the Knowledge of the Supreme Brahman in the hearts of His disciples; Who is Himself youthful but sits surrounded by old and great sages who are devoted to Brahman, the hands of the Supreme spiritual teacher is forming the Cin-Mudra (gesture of the Knowledge of Brahman) and Whose appearance is still and blissful, Who rejoices in His own Self which is reflected on His blissful face; Salutations to that Sri Dakshinamurthy. (Adi-Shankaracharya's Dakshinamurthy Stotram, Verse 1)
Silence is the english word for Mauna (मौन). The word Mauna comes from Muni, reference : मौन - Sanskrit Dictionary
According to the Viveka-Chudamoni, silence is the first gate to yoga:
yogasya prathamam dvaaram baang-nirodho (370). baang-nirodha means silence.
yogaanntaraayan mounena(7/15) :The yogi should win over his obstacles to yoga by adopting silence.
According to Linga-Purana also, mauna is a means of tapas or austerity
brahmacharyam tathaa maunam niraaharakatvam eva cha/ahimsaa sarvatah shantis tapa iti avidhiyate//
And the God of the Munis or Yogis and therefore of Silence is the Paramatma,Who is described as Shiva or Vishnu. In Svaatmaprakaashika we get
yat padam vimalam advayam shivam tat sad aham iti mounam ashraye --I take resort to silence to reach that Ever-unstinting Non-Dual Ever-existent Supreme Abode Shiva.
On the other hand, the famous dhyana-mantra of Sri Vishnu starting with 'saantaakaaram' also mentions Him as 'yogivir dhyana-gamyam' ie Whom the yogis attain by meditation.
So the God of Silence is the Paramatmaa, named as Shiva or Vishnu (or Surya or Durga/Kali or Ganesha etc).
Reference: Omkarnath Rachanavali, Mahamilan Math, Vol.1, page 375-379.