Have a look on the date_query parameter that has been added to WP 3.7
WP_Query#Date_Parameters and the author parameter.
Combine the two parameters as you need them to query all posts an author created in a given time:
$args = array(
'posts_per_page' = -1, // get all posts
'author' => get_the_author_meta( 'ID' ), // from this author ID
'date_query' => array( // in the last week
'year' => date('Y'),
'week' => date('W'),
'fields' => 'ids' // only return an array of post IDs
$results = new WP_Query( $args );
echo count( $results ); // display the number of results
Edit: I updated this answer based on input from @birgire to behave more performant.