Korea and India had relations from the middle of the first century AD when a princess of Ayodhya travelled to Korea and married there. Here is some more information on the relations between the two countries due to Buddhism.
Many Buddhist monks worked to shape the religion, a major early figure being Marananta, who came to Paekje in the 4th century A.D. However, it is the Samguksagi is not clear on how an Indian monk came to receive a warm welcome from the king. In A.D. 574 three Indian monks came to the peninsula with a Korean monk, Anhong, and initiated the construction of many monasteries and temples.
According to one story, King Ayuk (Muwa), identified with Ashoka Maurya of India, sent iron and gold to Korea to cast the image of the Buddha. Koreans used the metals to construct the monastery. However, the historical records show the Ashokan period was much earlier than the construction of the Hwangnyong Monastery.
After the introduction of Buddhism to Korea, many scholars and monks exhibited great enthusiasm for visiting India to learn more about Buddhism or for pilgrimages to places important to the memory of the Buddha. Some Korean monks set out for India in the early 6th century A.D.
The monk Kyomic was the first to visit India. He studied the Vinaya text, first going to the Samghana Temple of central India where he collected the Sanskrit text of the Mahisasaka Vinaya. Later an Indian monk, Devadatta (Pei-da-duo) ,came to Korea with Kyomic where he translated 72 books of the Vinaya under the patronage of the King Song of Paekje.
From the early 8th century onwards, Korean Buddhists showed a keen interest in India and Indian culture. The cultural bridge between the two countries grew from then onwards, Buddhist monks being credited for nurturing the relationship. Biographies of eminent monks of the Tang Dynasty in China recorded the brief histories of some 56 pilgrims who went to India. Among these was a Korean monk, A–nan-ya-bal-ma, who stayed at the Nalanda Monastery, possibly, even dying there. Another monk, Hye-op, also stayed in the Nalanda Monastery. One famous monk, Hyon-t’ae, (Sarvajnadeva) went to India via China, Tibet and Nepal. He went to Bodhgaya in Bihar and stayed near the Bodhi tree where the Buddha is said to have achieved enlightenment. In addition, Hyon-gaki (Parampujya) and Hsuan-chao also visited the Mahabodhi Monastery of Bihar.[Buddhism: The Bridge Between Korea and India]