The story of Lord Krishna teaching music to Sage Narada is described in Linga Purana, Volume 2, Chapter 3 but it doesn't mention Lord Krishna giving Vina to Sage Narada. The story goes like this: Sage Narada is a great devotee of Lord Vishnu, so he always wanted to please him. Because of this he used to sing for lord but he was ignored by Lord and other devas because there was another person Tumburu who was more efficient in music. So, Sage Narada decided to learn music and started severe austerities. Then the divine sound told him to go to Mansa lake and meet an Uluka naming Gyanabandhu. Narada went there and learned all type of music from Uluka then again goes to the abode of Lord Vishnu and sing there. There he has been told by Lord Vishnu that now he knows all about music but still he is not better than Tumburu. Lord Vishnu told him that when Lord Krishna will be incarnated he will teach you music and then you will become equal to Tumburu or even better than him. Then when Lord Krishna had taken birth, Sage Narada was taught music by Jambavanti, Satyabhama, Rukmini and at last by Lord Krishna. After that Lord Vishnu was pleased by his singing.
आहूय कृष्णो भगवान् स्वयमेव महामुनिम्।
अशिक्षयदमेयात्मा गानयोगमनुत्तमम्॥ १०३॥
Then lord Krsna, the incomprehensible soul, himself called the great sage and taught him the excellent art of music.
ततो ननर्त देवर्षिः प्रणिपत्य जनार्दनम्॥१०४॥
Thereafter, the sage Narada, surpassed the sage Tumburu. The divine sage Narada, thereafter, danced with joy, bowing to lord Visnu with respect.
उवाच च हृषीकेशः सर्वज्ञस्त्वं महामुने।
प्रहस्य गानयोगेन गायस्व मम सन्निधौ॥१०५॥
Lord Visnu then laughed and said- "O great sage, you have now attained the perfect knowledge. Having been equipped with the perfect knowledge pretty Well, you now sing in my presence.”