No, Shri Bhagwat Gita doesn't talk about caste system. It has reference only about Varnas and not Castes. Varna is the activity that one professes, this activity is called his karma. https://twitter.com/ambedkariteIND/status/1298086756789571584

Bhagwat Gita says that one's karma should be in line with his Dharma and that his Dharma should be in line with his Guna.

In 4/13 Shri Gita clearly says:

चातुर्वर्ण्यंमयासृष्टंगुणकर्मविभागशः।

The chaturvaranya (the four-class system) has been created on the basis of people’s vocations and merits and not on their birth and parentage.

In the 18th Chapter of the Bhagwat Gita, Shri Krishna gives information on the natural gunas of four varnas:

Peacefulness, self-control, austerity, purity, tolerance, honesty, knowledge, wisdom and Dharmikta —these are the natural qualities by which the brahmanas work.(18.42)

Heroism, power, determination, resourcefulness, courage in battle, generosity n leadership are the qualities of work for the ksatriyas.(18.43)

Farming, cow protection n business are the natural work for the vaisyas, and for the sudras there is labor and service to others.(18.44)

Dear H!ndus, Beware of Anti H!ndu Propagandists. Do your own research or read shastra instead of reading articles written by Anti H!ndus.