How do I print full date in the x axis of the line plot here?
For better control over the x-axis formatting, you can use the matplotlib.dates methods. In your case, MonthLocator and DateFormatter could be of interest. These can be used to adjust the x-axis as follows:
import matplotlib.dates as mdates

def time_series(start, end):
    time_series_df  = df.loc[(df['Date'] &gt;= start) &amp; (df['Date'] &lt;= end), ['Date', 'Value']]
    time_series_df = time_series_df.set_index('Date')

    fig, ax = plt.subplots(1)
    ax.plot(time_series_df)
    
    # Adjust the x-axis
    ax.xaxis.set_major_locator(mdates.MonthLocator(interval=3)) # Month intervals
    ax.xaxis.set_major_formatter(mdates.DateFormatter('%Y-%m-%d')) # date formatting
    # automaticall set font and rotation for date tick labels
    plt.gcf().autofmt_xdate()
    
    plt.xlabel('Time')
    plt.ylabel('PM2.5 Value')
    plt.title('PM2.5 Time Series')
    plt.show()

Note the interval=3 argument for MonthLocator. This is the frequency of the x-axis ticks.
Resulting plot:

Dataframe use for the plot above:
df = pd.DataFrame({'Date': pd.date_range(start='2014-01-01', end='2018-12-31'), 'Value': np.random.uniform(size=1826)})
df['Date'] = pd.to_datetime(df['Date'])