group by week in pandas

First, convert column date to_datetime and subtract one week as we want the sum for the week ahead of the date and not the week before that date.

Then use groupby with Grouper by W-MON and aggregate sum:

df['Date'] = pd.to_datetime(df['Date']) - pd.to_timedelta(7, unit='d')
df = df.groupby(['Name', pd.Grouper(key='Date', freq='W-MON')])['Quantity']
    .sum()
    .reset_index()
    .sort_values('Date')
print (df)

  Name       Date  Quantity
0   Apple 2017-07-10        90
3  orange 2017-07-10        20
1   Apple 2017-07-17        30
2  Orange 2017-07-24        40

Let's use groupby, resample with W-Mon, and sum:

df.groupby('Name').resample('W-Mon', on='Date').sum().reset_index().sort_values(by='Date')

Output:

     Name       Date  Quantity
0   Apple 2017-07-17        90
3  orange 2017-07-17        20
1   Apple 2017-07-24        30
2  Orange 2017-07-31        40