You can do that with a simple matrix product:
import pandas as pd
df = pd.DataFrame({
'Brand1': [1, 1, 1, 1, 1],
'Brand2': [0, 0, 0, 1, 1],
'Brand3': [0, 1, 1, 0, 0],
'Brand4': [1, 0, 0, 0, 1],
})
cross = df.T @ df
print(cross)
# Brand1 Brand2 Brand3 Brand4
# Brand1 5 2 2 2
# Brand2 2 2 0 1
# Brand3 2 0 2 0
# Brand4 2 1 0 2