Not without modification.
Open up class-challonge-widget.php in the challonge plugin folder.
Go to line 210 and add the following code after line 210:
.date_i18n(get_option( 'date_format' ) . ' ' . get_option( 'time_format' ),strtotime( $tourny->{ 'start-at' } )+ ( get_option( 'gmt_offset' ) * 3600 )).'<br />'
After that the code should look like this:
$ret .= '<br /><span class="challonge-info">'
.date_i18n(get_option( 'date_format' ) . ' ' . get_option( 'time_format' ),strtotime( $tourny->{ 'start-at' } )+ ( get_option( 'gmt_offset' ) * 3600 )).'<br />'
. esc_html( $lnk->participants_count ) . '/' . $lnk->signup_cap
. ' | ' . esc_html( ucwords( str_replace( '_', ' ', $tourny->state ) ) )
. '</span>';
$ret .= '</li>';
Bear in mind that this modification will be overwritten after a plugin update.